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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random variable</span></span>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert" typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<ul><li><a href="Probability" title="Probability">Probability</a>

<ul><li><a href="Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>

<li><a href="Determinism" title="Determinism">Determinism</a>

<ul><li><a href="Deterministic_system" title="Deterministic system">System</a></li></ul></li>

<li><a href="Indeterminism" title="Indeterminism">Indeterminism</a></li>

<li><a href="Randomness" title="Randomness">Randomness</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Probability_space" title="Probability space">Probability space</a></li>

<li><a href="Sample_space" title="Sample space">Sample space</a></li>

<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event</a>

<ul><li><a href="Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>

<li><a href="Elementary_event" title="Elementary event">Elementary event</a></li>

<li><a href="Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>

<li><a href="Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>

<li><a href="Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>

<ul><li><a href="Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a>

<ul><li><a href="Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>

<li><a href="Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>

<li><a href="Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>

<li><a href="Normal_distribution" title="Normal distribution">Normal distribution</a></li>

<li><a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>

<li><a href="Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>

<li><a href="Probability_measure" title="Probability measure">Probability measure</a></li>

<li>

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>

<li><a href="Expected_value" title="Expected value">Expected value</a></li>

<li><a href="Variance" title="Variance">Variance</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Realization_(probability)" title="Realization (probability)">Observed value</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a></li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Complementary_event" title="Complementary event">Complementary event</a></li>

<li><a href="Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>

<li><a href="Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>

<li><a href="Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>

<li><a href="Conditional_independence" title="Conditional independence">Conditional independence</a></li>

<li><a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>

<li><a href="Bayes'_theorem" title="Bayes' theorem">Bayes' theorem</a></li>

<li><a href="Boole's_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td>

</tr><tr><td class="sidebar-content">

<ul><li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li>

<li><a href="Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td>

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<p>A <b>random variable</b> (also called <b>random quantity</b>, <b>aleatory variable</b>, or <b>stochastic variable</b>) is a <a href="Mathematics" title="Mathematics"> mathematical</a> formalization of a quantity or object which depends on <a href="Randomness" title="Randomness">random</a> events.<sup id="cite_ref-:2_1-0" class="reference"><a href="#cite_note-:2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term 'random variable' in its mathematical definition refers to neither randomness nor variability<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> but instead is a mathematical <a href="Function_(mathematics)" title="Function (mathematics)">function</a> in which

</p>

<ul><li>the <a href="Domain_of_a_function" title="Domain of a function">domain</a> is the set of possible <a href="Outcome_(probability)" title="Outcome (probability)">outcomes</a> in a <a href="Sample_space" title="Sample space">sample space</a> (e.g. the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{H,T\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{H,T\}}</annotation>

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</math></span><img src="./f29ab87b5db9a3a422278eb0345cc491f00b47d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.059ex; height:2.843ex;" alt="{\displaystyle \{H,T\}}" loading="lazy"></span> which are the possible upper sides of a flipped coin heads <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> or tails <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> as the result from tossing a coin); and</li>

<li>the <a href="Range_of_a_function" title="Range of a function">range</a> is a <a href="Measurable_space" title="Measurable space">measurable space</a> (e.g. corresponding to the domain above, the range might be the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-1,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{-1,1\}}</annotation>

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</math></span><img src="./c0ffb8c7a09dad8456eee3669ee9a7e462fe3c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{-1,1\}}" loading="lazy"></span> if say heads <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> mapped to -1 and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> mapped to 1). Typically, the range of a random variable is a subset of the <a href="Real_number" title="Real number">real numbers</a>.</li></ul>

<p>Informally, randomness typically represents some fundamental element of chance, such as in the roll of a <a href="Dice" title="Dice">die</a>; it may also represent uncertainty, such as <a href="Measurement_error" class="mw-redirect" title="Measurement error">measurement error</a>.<sup id="cite_ref-:2_1-1" class="reference"><a href="#cite_note-:2-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, the <a href="Interpretation_of_probability" class="mw-redirect" title="Interpretation of probability">interpretation of probability</a> is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous <a href="Axiom" title="Axiom">axiomatic</a> setup.

</p><p>In the formal mathematical language of <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, a random variable is defined as a <a href="Measurable_function" title="Measurable function">measurable function</a> from a <a href="Probability_measure_space" class="mw-redirect" title="Probability measure space">probability measure space</a> (called the <i>sample space</i>) to a <a href="Measurable_space" title="Measurable space">measurable space</a>. This allows consideration of the <a href="Pushforward_measure" title="Pushforward measure">pushforward measure</a>, which is called the <i>distribution</i> of the random variable; the distribution is thus a <a href="Probability_measure" title="Probability measure">probability measure</a> on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be <a href="Independence_(probability_theory)" title="Independence (probability theory)">independent</a>.

</p><p>It is common to consider the special cases of <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a> and <a href="Probability_distribution#Absolutely_continuous_probability_distribution" title="Probability distribution">absolutely continuous random variables</a>, corresponding to whether a random variable is valued in a countable subset or in an interval of <a href="Real_number" title="Real number">real numbers</a>. There are other important possibilities, especially in the theory of <a href="Stochastic_process" title="Stochastic process">stochastic processes</a>, wherein it is natural to consider <a href="Random_sequence" title="Random sequence">random sequences</a> or <a href="Random_function" class="mw-redirect" title="Random function">random functions</a>. Sometimes a <i>random variable</i> is taken to be automatically valued in the real numbers, with more general random quantities instead being called <i><a href="Random_element" title="Random element">random elements</a></i>.

</p><p>According to <a href="George_Mackey" title="George Mackey">George Mackey</a>, <a href="Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a> was the first person "to think systematically in terms of random variables".<sup id="cite_ref-:3_3-0" class="reference"><a href="#cite_note-:3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A <b>random variable</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is a <a href="Measurable_function" title="Measurable function">measurable function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to E}">

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<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to E}</annotation>

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</math></span><img src="./5d99cb9d5a88548388921cc682df17876bdecaab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.082ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \to E}" loading="lazy"></span> from a sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> as a set of possible <a href="Outcome_(probability)" title="Outcome (probability)">outcomes</a> to a <a href="Measurable_space" title="Measurable space">measurable space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>. The technical axiomatic definition requires the sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> to belong to a <a href="Probability_space" title="Probability space">probability triple</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}</annotation>

</semantics>

</math></span><img src="./6d59d34e8fd75d8a0c739e420695cc695122073b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}" loading="lazy"></span> (see the <a href="#Measure-theoretic_definition">measure-theoretic definition</a>). A random variable is often denoted by capital <a href="Latin_script" title="Latin script">Roman letters</a> such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y,Z,T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,Y,Z,T}</annotation>

</semantics>

</math></span><img src="./0b048918c17de4e4a55d585170454117fe520b7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.172ex; height:2.509ex;" alt="{\displaystyle X,Y,Z,T}" loading="lazy"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>The probability that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> takes on a value in a measurable set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq E}</annotation>

</semantics>

</math></span><img src="./5fbdaaaadc900311922bfeaee0de2fa2bc6fe07a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.373ex; height:2.343ex;" alt="{\displaystyle S\subseteq E}" loading="lazy"></span> is written as

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}</annotation>

</semantics>

</math></span><img src="./13827ff57e01b13cceff9cf50cd9542cd4b7db70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.004ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Standard_case">Standard case</h3></div>

<p>In many cases, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is <a href="Real_number" title="Real number">real-valued</a>, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {R} }</annotation>

</semantics>

</math></span><img src="./403069620dea86b725997dd3b085172bd11f1bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="{\displaystyle E=\mathbb {R} }" loading="lazy"></span>. In some contexts, the term <a href="Random_element" title="Random element">random element</a> (see <a href="#Extensions">extensions</a>) is used to denote a random variable not of this form.

</p><p>When the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> (or range) of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is finite or <a href="Countable_set" title="Countable set">countably</a> infinite, the random variable is called a <b>discrete random variable</b><sup id="cite_ref-Yates_5-0" class="reference"><a href="#cite_note-Yates-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 399">: 399 </span></sup> and its distribution is a <a href="Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">discrete probability distribution</a>, i.e. can be described by a <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> that assigns a probability to each value in the image of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. If the image is uncountably infinite (usually an <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a>) then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is called a <b>continuous random variable</b>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In the special case that it is <a href="Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a>, its distribution can be described by a <a href="Probability_density_function" title="Probability density function">probability density function</a>, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p>Any random variable can be described by its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, which describes the probability that the random variable will be less than or equal to a certain value.

</p>

<div class="mw-heading mw-heading3"><h3 id="Extensions">Extensions</h3></div>

<p>The term "random variable" in statistics is traditionally limited to the <a href="Real_number" title="Real number">real-valued</a> case (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\mathbb {R} }</annotation>

</semantics>

</math></span><img src="./403069620dea86b725997dd3b085172bd11f1bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.552ex; height:2.176ex;" alt="{\displaystyle E=\mathbb {R} }" loading="lazy"></span>). In this case, the structure of the real numbers makes it possible to define quantities such as the <a href="Expected_value" title="Expected value">expected value</a> and <a href="Variance" title="Variance">variance</a> of a random variable, its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, and the <a href="Moment_(mathematics)" title="Moment (mathematics)">moments</a> of its distribution.

</p><p>However, the definition above is valid for any <a href="Measurable_space" title="Measurable space">measurable space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> of values. Thus one can consider random elements of other sets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>, such as random <a href="Boolean-valued_function" title="Boolean-valued function">Boolean values</a>, <a href="Categorical_variable" title="Categorical variable">categorical values</a>, <a href="Covariance_matrix#Complex_random_vectors" title="Covariance matrix">complex numbers</a>, <a href="Random_vector" class="mw-redirect" title="Random vector">vectors</a>, <a href="Random_matrix" title="Random matrix">matrices</a>, <a href="Random_sequence" title="Random sequence">sequences</a>, <a href="Tree_(graph_theory)" title="Tree (graph theory)">trees</a>, <a href="Random_compact_set" title="Random compact set">sets</a>, <a href="Shape" title="Shape">shapes</a>, <a href="Manifold" title="Manifold">manifolds</a>, and <a href="Random_function" class="mw-redirect" title="Random function">functions</a>. One may then specifically refer to a <i>random variable of <a href="Data_type" title="Data type">type</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span></i>, or an <i><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>-valued random variable</i>.

</p><p>This more general concept of a <a href="Random_element" title="Random element">random element</a> is particularly useful in disciplines such as <a href="Graph_theory" title="Graph theory">graph theory</a>, <a href="Machine_learning" title="Machine learning">machine learning</a>, <a href="Natural_language_processing" title="Natural language processing">natural language processing</a>, and other fields in <a href="Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a>, where one is often interested in modeling the random variation of non-numerical <a href="Data_structure" title="Data structure">data structures</a>. In some cases, it is nonetheless convenient to represent each element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>, using one or more real numbers. In this case, a random element may optionally be represented as a <a href="Random_vector" class="mw-redirect" title="Random vector">vector of real-valued random variables</a> (all defined on the same underlying probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, which allows the different random variables to <a href="Mutual_information" title="Mutual information">covary</a>). For example:

</p>

<ul><li>A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1\ 0\ 0\ 0\ \cdots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1\ 0\ 0\ 0\ \cdots )}</annotation>

</semantics>

</math></span><img src="./8f6e1c3d8f13ec2d8be9bca0eb66a8558367c2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (1\ 0\ 0\ 0\ \cdots )}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\ 1\ 0\ 0\ \cdots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0\ 1\ 0\ 0\ \cdots )}</annotation>

</semantics>

</math></span><img src="./691482810fe3ec32b87adaa3f47c05fccc580662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (0\ 1\ 0\ 0\ \cdots )}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0\ 0\ 1\ 0\ \cdots )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0\ 0\ 1\ 0\ \cdots )}</annotation>

</semantics>

</math></span><img src="./ee47afca946862178e185c881db6e45df5954927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.843ex;" alt="{\displaystyle (0\ 0\ 1\ 0\ \cdots )}" loading="lazy"></span> and the position of the 1 indicates the word.</li>

<li>A random sentence of given length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> may be represented as a vector of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> random words.</li>

<li>A <a href="Random_graph" title="Random graph">random graph</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> given vertices may be represented as a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\times N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation>

</semantics>

</math></span><img src="./99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="{\displaystyle N\times N}" loading="lazy"></span> matrix of random variables, whose values specify the <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the random graph.</li>

<li>A <a href="Random_function" class="mw-redirect" title="Random function">random function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> may be represented as a collection of random variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>

</semantics>

</math></span><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"></span>, giving the function's values at the various points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in the function's domain. The <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation>

</semantics>

</math></span><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"></span> are ordinary real-valued random variables provided that the function is real-valued. For example, a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> is a random function of time, a <a href="Random_vector" class="mw-redirect" title="Random vector">random vector</a> is a random function of some <a href="Index_set" title="Index set">index set</a> such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,\ldots ,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1,2,\ldots ,n}</annotation>

</semantics>

</math></span><img src="./e3e6b15e92183431bb62b787fcdcbdcbe8b40234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.932ex; height:2.509ex;" alt="{\displaystyle 1,2,\ldots ,n}" loading="lazy"></span>, and <a href="Random_field" title="Random field">random field</a> is a random function on any set (typically time, space, or a discrete set).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Distribution_functions">Distribution functions</h2></div>

<p>If a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./b68ecb50490003c72b7d6627145965f602b302b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.984ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \to \mathbb {R} }" loading="lazy"></span> defined on the probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}</annotation>

</semantics>

</math></span><img src="./6d59d34e8fd75d8a0c739e420695cc695122073b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}" loading="lazy"></span> is given, we can ask questions like "How likely is it that the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is equal to 2?". This is the same as the probability of the event <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega :X(\omega )=2\}\,\!}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )=2\}\,\!}</annotation>

</semantics>

</math></span><img src="./4cb4fe1d84c37c35452c6c5721382a735981d434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:15.591ex; height:2.843ex;" alt="{\displaystyle \{\omega :X(\omega )=2\}\,\!}" loading="lazy"></span> which is often written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=2)\,\!}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(X=2)\,\!}</annotation>

</semantics>

</math></span><img src="./c7e44de4aa10f40c4de59aa17c6c607cb3a61e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.183ex; height:2.843ex;" alt="{\displaystyle P(X=2)\,\!}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}(2)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}(2)}</annotation>

</semantics>

</math></span><img src="./5188e81eeb129348060f1b646de0f78570b11539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.863ex; height:2.843ex;" alt="{\displaystyle p_{X}(2)}" loading="lazy"></span> for short.

</p><p>Recording all these probabilities of outputs of a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> yields the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. The probability distribution "forgets" about the particular probability space used to define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> and only records the probabilities of various output values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Such a probability distribution, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is real-valued, can always be captured by its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}</annotation>

</semantics>

</math></span><img src="./f81c05aba576a12b4e05ee3f4cba709dd16139c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.165ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}" loading="lazy"></span></dd></dl>

<p>and sometimes also using a <a href="Probability_density_function" title="Probability density function">probability density function</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}}</annotation>

</semantics>

</math></span><img src="./17fd6605a04f97c6bedb0a9632f9f023cb18dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.772ex; height:2.509ex;" alt="{\displaystyle f_{X}}" loading="lazy"></span>. In <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure-theoretic</a> terms, we use the random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> to "push-forward" the measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> to a measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>

</semantics>

</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>. The measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>

</semantics>

</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> is called the "(probability) distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>" or the "law of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>".

<sup id="cite_ref-:Billingsley_9-0" class="reference"><a href="#cite_note-:Billingsley-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The density <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}=dp_{X}/d\mu }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}=dp_{X}/d\mu }</annotation>

</semantics>

</math></span><img src="./a44b7ef2a2264624ff3718feac3ada197f5bd465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.668ex; height:2.843ex;" alt="{\displaystyle f_{X}=dp_{X}/d\mu }" loading="lazy"></span>, the <a href="Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}}</annotation>

</semantics>

</math></span><img src="./4bc900e2770ed796e420eb5aa0852193a1919ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.891ex; height:2.009ex;" alt="{\displaystyle p_{X}}" loading="lazy"></span> with respect to some reference measure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> (often, this reference measure is the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> in the case of continuous random variables, or the <a href="Counting_measure" title="Counting measure">counting measure</a> in the case of discrete random variables).

The underlying probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as <a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">correlation and dependence</a> or <a href="Independence_(probability_theory)" title="Independence (probability theory)">independence</a> based on a <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a> of two or more random variables on the same probability space. In practice, one often disposes of the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> altogether and just puts a measure on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on <a href="Quantile_function" title="Quantile function">quantile functions</a> for fuller development.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Discrete_random_variable">Discrete random variable</h3></div>

<p>Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.

</p><p>Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }</annotation>

</semantics>

</math></span><img src="./d91f95b28c52ac8f55c312af264b51ba3bf7b2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.855ex; height:2.843ex;" alt="{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }" loading="lazy"></span>.

</p><p>In examples such as these, the <a href="Sample_space" title="Sample space">sample space</a> is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{a_{n}\},\{b_{n}\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \{a_{n}\},\{b_{n}\}}</annotation>

</semantics>

</math></span><img src="./82803ec71082ae4fdb9e9821758578b27d8a83d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.348ex; height:2.843ex;" alt="{\textstyle \{a_{n}\},\{b_{n}\}}" loading="lazy"></span> are countable sets of real numbers, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b_{n}>0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle b_{n}>0}</annotation>

</semantics>

</math></span><img src="./eb23ccb34be64412e4d74b40cae449bf0b7ff1fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.477ex; height:2.509ex;" alt="{\textstyle b_{n}>0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n}b_{n}=1}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{n}b_{n}=1}</annotation>

</semantics>

</math></span><img src="./91ae5ee7ad35a0fdbafb8ffb776e32f20ca32238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.536ex; height:3.009ex;" alt="{\textstyle \sum _{n}b_{n}=1}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)}</annotation>

</semantics>

</math></span><img src="./6becdc99a562cff63872057cfdd2354236b8acd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.353ex; height:3.009ex;" alt="{\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)}" loading="lazy"></span> is a discrete distribution function. Here <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{t}(x)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{t}(x)=0}</annotation>

</semantics>

</math></span><img src="./d9b5c97ea938411c14762eff44de74add8a21f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.258ex; height:2.843ex;" alt="{\displaystyle \delta _{t}(x)=0}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<t}</annotation>

</semantics>

</math></span><img src="./29d1dc488d4ab095183b700be244e98632e0b5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.009ex;" alt="{\displaystyle x<t}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{t}(x)=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{t}(x)=1}</annotation>

</semantics>

</math></span><img src="./458404a2acdb99c795f12f1bd9c5cb3d516b2efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.258ex; height:2.843ex;" alt="{\displaystyle \delta _{t}(x)=1}" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\geq t}</annotation>

</semantics>

</math></span><img src="./0a02705731d972f028007efcdb012b3ac80f954e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle x\geq t}" loading="lazy"></span>. Taking for instance an enumeration of all rational numbers as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a_{n}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a_{n}\}}</annotation>

</semantics>

</math></span><img src="./339d461b497fb8f8670bb29308fe09f0e7bfd34a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.773ex; height:2.843ex;" alt="{\displaystyle \{a_{n}\}}" loading="lazy"></span> , one gets a discrete function that is not necessarily a <a href="Step_function" title="Step function">step function</a> (<a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> constant).

</p>

<p>The possible outcomes for one coin toss can be described by the sample space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}">

<mtext>heads</mtext>

</mrow>

<mtext>tails</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}</annotation>

</semantics>

</math></span><img src="./408c39b68acbad53a0195285b2dee7f5978bd967.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.109ex; height:2.843ex;" alt="{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}" loading="lazy"></span>. We can introduce a real-valued random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> that models a $1 payoff for a successful bet on heads as follows:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<mtext>heads</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

<mtext>tails</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}</annotation>

</semantics>

</math></span></span>

</p><p>If the coin is a <a href="Fair_coin" title="Fair coin">fair coin</a>, <i>Y</i> has a <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}}</annotation>

</semantics>

</math></span><img src="./6e8b3fed56c8af1f38961f6e4ec0d64fe50ecb4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.625ex; height:2.509ex;" alt="{\displaystyle f_{Y}}" loading="lazy"></span> given by:

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}}</annotation>

</semantics>

</math></span></span>

</p>

<p>A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers <i>n</i><sub>1</sub> and <i>n</i><sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable <i>X</i> given by the function that maps the pair to the sum:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}}</annotation>

</semantics>

</math></span></span>

and (if the dice are <a href="Fair_die" class="mw-redirect" title="Fair die">fair</a>) has a probability mass function <i>f</i><sub><i>X</i></sub> given by:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Continuous_random_variable">Continuous random variable</h3></div>

<p>Formally, a continuous random variable is a random variable whose <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is <a href="Continuous_function" title="Continuous function">continuous</a> everywhere.<sup id="cite_ref-:0_10-0" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> There are no "<a href="Discontinuity_(mathematics)" class="mw-redirect" title="Discontinuity (mathematics)">gaps</a>", which would correspond to numbers which have a finite probability of <a href="Outcome_(probability)" title="Outcome (probability)">occurring</a>. Instead, continuous random variables <a href="Almost_never" class="mw-redirect" title="Almost never">almost never</a> take an exact prescribed value <i>c</i> (formally, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}</annotation>

</semantics>

</math></span><img src="./42ac5d4ef6961bbacd15206d924480ad80a62943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.05ex; height:2.843ex;" alt="{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}" loading="lazy"></span>) but there is a positive probability that its value will lie in particular <a href="Interval_(mathematics)" title="Interval (mathematics)">intervals</a> which can be <a href="Arbitrarily_small" class="mw-redirect" title="Arbitrarily small">arbitrarily small</a>. Continuous random variables usually admit <a href="Probability_density_function" title="Probability density function">probability density functions</a> (PDF), which characterize their CDF and <a href="Probability_measure" title="Probability measure">probability measures</a>;

such distributions are also called <a href="Absolutely_continuous_random_variable" class="mw-redirect" title="Absolutely continuous random variable">absolutely continuous</a>; but some continuous distributions are <a href="Singular_distribution" title="Singular distribution">singular</a>, or mixes of an absolutely continuous part and a singular part.

</p><p>An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, <i><b>X</b></i> = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any <i>range</i> of values. For example, the probability of choosing a number in [0, 180] is <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. Instead of speaking of a probability mass function, we say that the probability <i>density</i> of <i><b>X</b></i> is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.

</p><p>More formally, given any <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}</annotation>

</semantics>

</math></span><img src="./6a291b85e1772a1749a65c841769efcec2931376.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.788ex; height:2.843ex;" alt="{\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}" loading="lazy"></span>, a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}</annotation>

</semantics>

</math></span><img src="./e796af5f27ecaec3fdf010383da162a4d29ea04d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.204ex; height:2.843ex;" alt="{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}" loading="lazy"></span> is called a "<a href="Continuous_uniform_distribution" title="Continuous uniform distribution">continuous uniform</a> random variable" (CURV) if the probability that it takes a value in a <a href="Subinterval" class="mw-redirect" title="Subinterval">subinterval</a> depends only on the length of the subinterval. This implies that the probability of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{I}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{I}}</annotation>

</semantics>

</math></span><img src="./064414454fbbf8ad5a6132839c02cf853b1049f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle X_{I}}" loading="lazy"></span> falling in any subinterval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [c,d]\subseteq [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [c,d]\subseteq [a,b]}</annotation>

</semantics>

</math></span><img src="./32c4b90060cfad2522da60caeab608de43226f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.204ex; height:2.843ex;" alt="{\displaystyle [c,d]\subseteq [a,b]}" loading="lazy"></span> is <a href="Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the <a href="Lebesgue_measure" title="Lebesgue measure">length</a> of the subinterval, that is, if <span class="texhtml"><i>a</i> ≤ <i>c</i> ≤ <i>d</i> ≤ <i>b</i></span>, one has

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}}</annotation>

</semantics>

</math></span></span>

</p><p>where the last equality results from the <a href="Probability_axioms#Unitarity" title="Probability axioms">unitarity axiom</a> of probability. The <a href="Probability_density_function" title="Probability density function">probability density function</a> of a CURV <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {U} [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {U} [a,b]}</annotation>

</semantics>

</math></span><img src="./8a2c5c9c387a71f6b8511c8360740aed05476755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.377ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {U} [a,b]}" loading="lazy"></span> is given by the <a href="Indicator_function" title="Indicator function">indicator function</a> of its interval of <a href="Support_(mathematics)" title="Support (mathematics)">support</a> normalized by the interval's length: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}">

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>otherwise</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}</annotation>

</semantics>

</math></span></span>Of particular interest is the uniform distribution on the <a href="Unit_interval" title="Unit interval">unit interval</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math></span><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"></span>. Samples of any desired <a href="Probability_distribution" title="Probability distribution">probability distribution</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {D} }</annotation>

</semantics>

</math></span><img src="./70a9dd309ae777cd525cdf07df0e2b132a8fe6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \operatorname {D} }" loading="lazy"></span> can be generated by calculating the <a href="Quantile_function" title="Quantile function">quantile function</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {D} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {D} }</annotation>

</semantics>

</math></span><img src="./70a9dd309ae777cd525cdf07df0e2b132a8fe6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \operatorname {D} }" loading="lazy"></span> on a <a href="Random_number_generation" title="Random number generation">randomly-generated number</a> distributed uniformly on the unit interval. This exploits <a href="Cumulative_distribution_function#Properties" title="Cumulative distribution function">properties of cumulative distribution functions</a>, which are a unifying framework for all random variables.

</p>

<div class="mw-heading mw-heading3"><h3 id="Mixed_type">Mixed type</h3></div>

<p>A <b>mixed random variable</b> is a random variable whose <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is neither <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete</a> nor <a href="Continuous_function" title="Continuous function">everywhere-continuous</a>.<sup id="cite_ref-:0_10-1" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> It can be realized as a mixture of a discrete random variable and a continuous random variable; in which case the <abbr title="cumulative distribution function">CDF</abbr> will be the weighted average of the CDFs of the component variables.<sup id="cite_ref-:0_10-2" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, <i><b>X</b></i> = −1; otherwise <i><b>X</b></i> = the value of the spinner as in the preceding example. There is a probability of <span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.

</p><p>Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see <a href="Lebesgue's_decomposition_theorem#Refinement" title="Lebesgue's decomposition theorem">Lebesgue's decomposition theorem § Refinement</a>. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).

</p>

<div class="mw-heading mw-heading2"><h2 id="Measure-theoretic_definition">Measure-theoretic definition</h2></div>

<p>The most formal, <a href="Axiomatic" class="mw-redirect" title="Axiomatic">axiomatic</a> definition of a random variable involves <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>. Continuous random variables are defined in terms of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the <a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a <a href="Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the <a href="Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a>, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of <a href="Union_(set_theory)" title="Union (set theory)">unions</a> and/or <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersections</a> of such intervals.<sup id="cite_ref-UCSB_11-0" class="reference"><a href="#cite_note-UCSB-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p><p>The measure-theoretic definition is as follows.

</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> be a <a href="Probability_space" title="Probability space">probability space</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy"></span> a <a href="Measurable_space" title="Measurable space">measurable space</a>. Then an <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy"></span>-valued random variable</b> is a measurable function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to E}</annotation>

</semantics>

</math></span><img src="./5d99cb9d5a88548388921cc682df17876bdecaab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.082ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \to E}" loading="lazy"></span>, which means that, for every subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./81710349199c3fc6bcbe14f6758e0c2f1fe9bbca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {E}}}" loading="lazy"></span>, its <a href="Preimage" class="mw-redirect" title="Preimage">preimage</a> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>-measurable; <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}(B)\in {\mathcal {F}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{-1}(B)\in {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./9bed6cfd3c53e4c0affd22f625651158ec8637b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.67ex; height:3.176ex;" alt="{\displaystyle X^{-1}(B)\in {\mathcal {F}}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}</annotation>

</semantics>

</math></span><img src="./daed213f811c2e8cc77dbb56297b2f8b4688a625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.549ex; height:3.176ex;" alt="{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}" loading="lazy"></span>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This definition enables us to measure any subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./81710349199c3fc6bcbe14f6758e0c2f1fe9bbca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {E}}}" loading="lazy"></span> in the target space by looking at its preimage, which by assumption is measurable.

</p><p>In more intuitive terms, a member of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is a possible outcome, a member of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is a measurable subset of possible outcomes, the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> gives the probability of each such measurable subset, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> represents the set of values that the random variable can take (such as the set of real numbers), and a member of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"></span> is a "well-behaved" (measurable) subset of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.

</p><p>When <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> is a <a href="Topological_space" title="Topological space">topological space</a>, then the most common choice for the <a href="%CE%A3-algebra" title="Σ-algebra">σ-algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math></span><img src="./9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"></span> is the <a href="Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(E)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(E)}</annotation>

</semantics>

</math></span><img src="./3164c0c1413cc8c9b0b9d8b7052d8c7173699f9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(E)}" loading="lazy"></span>, which is the σ-algebra generated by the collection of all open sets in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>. In such case the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

</semantics>

</math></span><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy"></span>-valued random variable is called an <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>-valued random variable</b>. Moreover, when the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> is the real line <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>, then such a real-valued random variable is called simply a <b>random variable</b>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Real-valued_random_variables">Real-valued random variables</h3></div>

<p>In this case the observation space is the set of real numbers. Recall, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> is the probability space. For a real observation space, the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./0e26dcc35ac81a12c1263c86b05225ef113089fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.984ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }" loading="lazy"></span> is a real-valued random variable if

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}</annotation>

</semantics>

</math></span><img src="./117cd0be4499c834fefa587d5ec055a78f47d178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.009ex; height:2.843ex;" alt="{\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}" loading="lazy"></span></dd></dl>

<p>This definition is a special case of the above because the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}</annotation>

</semantics>

</math></span><img src="./70d40dbc8dd41bbe90d6242042f72e62bafea8f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.595ex; height:2.843ex;" alt="{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}" loading="lazy"></span> generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}</annotation>

</semantics>

</math></span><img src="./967b79350e615a40cee0dd0102fee55bfb3c5d3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.093ex; height:3.176ex;" alt="{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Moments">Moments</h2></div>

<p>The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of <a href="Expected_value" title="Expected value">expected value</a> of a random variable, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation>

</semantics>

</math></span><img src="./44dd294aa33c0865f58e2b1bdaf44ebe911dbf93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]}" loading="lazy"></span>, and also called the <b>first <a href="Moment_(mathematics)" title="Moment (mathematics)">moment</a>.</b> In general, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [f(X)]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [f(X)]}</annotation>

</semantics>

</math></span><img src="./c407e0dfff7f7d09b8a81f9ccc2f078bffa783ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.944ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [f(X)]}" loading="lazy"></span> is not equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\operatorname {E} [X])}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\operatorname {E} [X])}</annotation>

</semantics>

</math></span><img src="./358c53d63b891b58814383d8beba46f69695632f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.944ex; height:2.843ex;" alt="{\displaystyle f(\operatorname {E} [X])}" loading="lazy"></span>. Once the "average value" is known, one could then ask how far from this average value the values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> typically are, a question that is answered by the <a href="Variance" title="Variance">variance</a> and <a href="Standard_deviation" title="Standard deviation">standard deviation</a> of a random variable. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation>

</semantics>

</math></span><img src="./44dd294aa33c0865f58e2b1bdaf44ebe911dbf93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]}" loading="lazy"></span> can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.

</p><p>Mathematically, this is known as the (generalised) <a href="Problem_of_moments" class="mw-redirect" title="Problem of moments">problem of moments</a>: for a given class of random variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, find a collection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{i}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f_{i}\}}</annotation>

</semantics>

</math></span><img src="./a98b80e62d2a39bb2bdf0dc2c882a3fd4b810c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.264ex; height:2.843ex;" alt="{\displaystyle \{f_{i}\}}" loading="lazy"></span> of functions such that the expectation values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [f_{i}(X)]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [f_{i}(X)]}</annotation>

</semantics>

</math></span><img src="./0e4e5f5f0c5d751d4d1bf63dea54ff9765683a53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.605ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [f_{i}(X)]}" loading="lazy"></span> fully characterise the <a href="Probability_distribution" title="Probability distribution">distribution</a> of the random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.

</p><p>Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(X)=X}</annotation>

</semantics>

</math></span><img src="./b68563bf25d5ecb2a4dcc2769577eeab1e1ab955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.146ex; height:2.843ex;" alt="{\displaystyle f(X)=X}" loading="lazy"></span> of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a <a href="Categorical_variable" title="Categorical variable">categorical</a> random variable <i>X</i> that can take on the <a href="Nominal_data" class="mw-redirect" title="Nominal data">nominal</a> values "red", "blue" or "green", the real-valued function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X={\text{green}}]}">

<mtext>green</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X={\text{green}}]}</annotation>

</semantics>

</math></span><img src="./e41a3122d8561d29d90be48b6c1fb0f94d8e2a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.804ex; height:2.843ex;" alt="{\displaystyle [X={\text{green}}]}" loading="lazy"></span> can be constructed; this uses the <a href="Iverson_bracket" title="Iverson bracket">Iverson bracket</a>, and has the value 1 if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> has the value "green", 0 otherwise. Then, the <a href="Expected_value" title="Expected value">expected value</a> and other moments of this function can be determined.

</p>

<div class="mw-heading mw-heading2"><h2 id="Functions_of_random_variables">Functions of random variables</h2></div>

<p>A new random variable <i>Y</i> can be defined by <a href="Function_composition" title="Function composition">applying</a> a real <a href="Measurable_function" title="Measurable function">Borel measurable function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./4ae6a1665073bd67dd6b4ef31d23fedad6c62c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.12ex; height:2.509ex;" alt="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }" loading="lazy"></span> to the outcomes of a <a href="Real-valued" class="mw-redirect" title="Real-valued">real-valued</a> random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. That is, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=g(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=g(X)}</annotation>

</semantics>

</math></span><img src="./84fd834147d6175f13477acff6567727265d0000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.777ex; height:2.843ex;" alt="{\displaystyle Y=g(X)}" loading="lazy"></span>. The <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> is then

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}</annotation>

</semantics>

</math></span><img src="./f34db583acc468a9227b7a3ef11a8967480f7d75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.242ex; height:2.843ex;" alt="{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}" loading="lazy"></span></dd></dl>

<p>If function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is invertible (i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=g^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h=g^{-1}}</annotation>

</semantics>

</math></span><img src="./224c8e57f10e59865f80b60f807aa160f988792f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.888ex; height:3.009ex;" alt="{\displaystyle h=g^{-1}}" loading="lazy"></span> exists, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>'s <a href="Inverse_function" title="Inverse function">inverse function</a>) and is either <a href="Monotonic_function" title="Monotonic function">increasing or decreasing</a>, then the previous relation can be extended to obtain

</p>

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

<mtext> increasing</mtext>

</mrow>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

<mtext> decreasing</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}</annotation>

</semantics>

</math></span><img src="./fda82f7b493f5a2a742ad99a7e5b3e23916b88cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:81.133ex; height:8.843ex;" alt="{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}" loading="lazy"></span></dd></dl>

<p>With the same hypotheses of invertibility of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>, assuming also <a href="Differentiability" class="mw-redirect" title="Differentiability">differentiability</a>, the relation between the <a href="Probability_density_function" title="Probability density function">probability density functions</a> can be found by differentiating both sides of the above expression with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>, in order to obtain<sup id="cite_ref-:0_10-3" class="reference"><a href="#cite_note-:0-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}</annotation>

</semantics>

</math></span><img src="./e26c11c24bf65c0792d0abfc5eb9039a9f8a2dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.964ex; height:6.509ex;" alt="{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}" loading="lazy"></span></dd></dl>

<p>If there is no invertibility of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> but each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> admits at most a countable number of roots (i.e., a finite, or countably infinite, number of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g(x_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=g(x_{i})}</annotation>

</semantics>

</math></span><img src="./4e8c8917fac0255b472f3fcbb3dfd5349ba2314e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.309ex; height:2.843ex;" alt="{\displaystyle y=g(x_{i})}" loading="lazy"></span>) then the previous relation between the <a href="Probability_density_function" title="Probability density function">probability density functions</a> can be generalized with

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|}</annotation>

</semantics>

</math></span><img src="./86def9876ca12d7e6c6e073b7b9c408b9ed44c31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.576ex; height:7.176ex;" alt="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=g_{i}^{-1}(y)}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=g_{i}^{-1}(y)}</annotation>

</semantics>

</math></span><img src="./2ae01cf9051bbaa9c5b4e78477edf51930889e6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.644ex; height:3.343ex;" alt="{\displaystyle x_{i}=g_{i}^{-1}(y)}" loading="lazy"></span>, according to the <a href="Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>. The formulas for densities do not demand <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> to be increasing.

</p><p>In the measure-theoretic, <a href="Probability_axioms" title="Probability axioms">axiomatic approach</a> to probability, if a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> and a <a href="Measurable_function" title="Measurable function">Borel measurable function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./4ae6a1665073bd67dd6b4ef31d23fedad6c62c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.12ex; height:2.509ex;" alt="{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=g(X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=g(X)}</annotation>

</semantics>

</math></span><img src="./84fd834147d6175f13477acff6567727265d0000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.777ex; height:2.843ex;" alt="{\displaystyle Y=g(X)}" loading="lazy"></span> is also a random variable on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span>, since the composition of measurable functions <a href="Closure_(mathematics)" title="Closure (mathematics)">is also measurable</a>. (However, this is not necessarily true if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> is <a href="Lebesgue_measurable" class="mw-redirect" title="Lebesgue measurable">Lebesgue measurable</a>.) The same procedure that allowed one to go from a probability space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,P)}</annotation>

</semantics>

</math></span><img src="./c9c592eb392b832a4e583bbde12cffe5fb5cf975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.267ex; height:2.843ex;" alt="{\displaystyle (\Omega ,P)}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,dF_{X})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,dF_{X})}</annotation>

</semantics>

</math></span><img src="./c7ddffbe92024b04bcde1bad6b9468442be4851b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.864ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,dF_{X})}" loading="lazy"></span> can be used to obtain the distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example_1">Example 1</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> be a real-valued, <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a> and let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}}</annotation>

</semantics>

</math></span><img src="./164fd6721c56072e02ca8627ae6c18b1efa64e71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.923ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}}" loading="lazy"></span>.

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}</annotation>

</semantics>

</math></span><img src="./bd95dc6913b00a023f868356b416e9894073499b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.388ex; height:3.176ex;" alt="{\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y<0}</annotation>

</semantics>

</math></span><img src="./dd6a57e5eb282c81c2bb6f5e313012fa77bc08a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y<0}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X^{2}\leq y)=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(X^{2}\leq y)=0}</annotation>

</semantics>

</math></span><img src="./24970bfd4938821ee236de96780146c105db2be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.121ex; height:3.176ex;" alt="{\displaystyle P(X^{2}\leq y)=0}" loading="lazy"></span>, so

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}</annotation>

</semantics>

</math></span><img src="./bcdceed3c03d3e2224bd086e1cf5a5866a17d611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.596ex; height:2.843ex;" alt="{\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\geq 0}</annotation>

</semantics>

</math></span><img src="./130e8795bc869a5b823133c5a0972693605c00bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y\geq 0}" loading="lazy"></span>, then

</p>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}</annotation>

</semantics>

</math></span><img src="./647f3a59502b685931f784766844a78fea5645e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:49.957ex; height:3.509ex;" alt="{\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}" loading="lazy"></span></dd></dl>

<p>so

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}</annotation>

</semantics>

</math></span><img src="./5a4965737980580e870cc77fa21d507b0a13b5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.137ex; height:3.176ex;" alt="{\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Example_2">Example 2</h3></div>

<p>Suppose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is a random variable with a cumulative distribution

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}}</annotation>

</semantics>

</math></span><img src="./c71de30d81204fddf41a7ab8d92bab0e1997d63a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.612ex; height:6.009ex;" alt="{\displaystyle F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta >0}</annotation>

</semantics>

</math></span><img src="./0b0ac07626379d065418cc158ce6be9aeccf33b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta >0}" loading="lazy"></span> is a fixed parameter. Consider the random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\mathrm {log} (1+e^{-X}).}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=\mathrm {log} (1+e^{-X}).}</annotation>

</semantics>

</math></span><img src="./15eedf0c5e52e3d164e9e67c935cfd91f945fb15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.297ex; height:3.176ex;" alt="{\displaystyle Y=\mathrm {log} (1+e^{-X}).}" loading="lazy"></span> Then,

</p>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}</annotation>

</semantics>

</math></span><img src="./b546da06b42000fefd3249d4091832619aa90f52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:67.802ex; height:3.176ex;" alt="{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}" loading="lazy"></span></dd></dl>

<p>The last expression can be calculated in terms of the cumulative distribution of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math></span><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"></span> so

</p>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./f4909911022c68c07e09a3cd4722e9c60b62a4f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:31.846ex; height:22.343ex;" alt="{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>which is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) of an <a href="Exponential_distribution" title="Exponential distribution">exponential distribution</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example_3">Example 3</h3></div>

<p>Suppose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is a random variable with a <a href="Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a>, whose density is

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.}</annotation>

</semantics>

</math></span><img src="./8b083a3c3826d10f695ca1b3e66ba8abb567165a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.932ex; height:6.176ex;" alt="{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.}" loading="lazy"></span></dd></dl>

<p>Consider the random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}.}</annotation>

</semantics>

</math></span><img src="./5d8194b25626ef14cd837717548e36d3ffebcf2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.57ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}.}" loading="lazy"></span> We can find the density using the above formula for a change of variables:

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}</annotation>

</semantics>

</math></span><img src="./2759b7e26c5b7f286c6ecadec6dbfa550a5400f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.61ex; height:7.176ex;" alt="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}" loading="lazy"></span></dd></dl>

<p>In this case the change is not <a href="Monotonic_function" title="Monotonic function">monotonic</a>, because every value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> has two corresponding values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}</annotation>

</semantics>

</math></span><img src="./57f4674302af1273e9ca28d3115ec5e9d91fcc16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.03ex; height:6.843ex;" alt="{\displaystyle f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}" loading="lazy"></span></dd></dl>

<p>The inverse transformation is

</p>

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=g^{-1}(y)={\sqrt {y}}}</annotation>

</semantics>

</math></span><img src="./cfa3b24dd2edf00cfbf9641e51b548e0cebd85aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.034ex; height:3.509ex;" alt="{\displaystyle x=g^{-1}(y)={\sqrt {y}}}" loading="lazy"></span></dd></dl>

<p>and its derivative is

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.}</annotation>

</semantics>

</math></span><img src="./c65d4d846f8c011ac2a8021767702285647cd7d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.303ex; height:6.843ex;" alt="{\displaystyle {\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.}" loading="lazy"></span></dd></dl>

<p>Then,

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.}</annotation>

</semantics>

</math></span><img src="./505d29170afe32fcb1823bfffdde7bf35a59e52a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.873ex; height:6.509ex;" alt="{\displaystyle f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.}" loading="lazy"></span></dd></dl>

<p>This is a <a href="Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with one <a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degree of freedom</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example_4">Example 4</h3></div>

<p>Suppose <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is a random variable with a <a href="Normal_distribution" title="Normal distribution">normal distribution</a>, whose density is

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}</annotation>

</semantics>

</math></span><img src="./48359f16b3ba33c0bc3cfe7a70bddd9d1e88c7fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.918ex; height:6.176ex;" alt="{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}" loading="lazy"></span></dd></dl>

<p>Consider the random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=X^{2}.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=X^{2}.}</annotation>

</semantics>

</math></span><img src="./5d8194b25626ef14cd837717548e36d3ffebcf2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.57ex; height:2.676ex;" alt="{\displaystyle Y=X^{2}.}" loading="lazy"></span> We can find the density using the above formula for a change of variables:

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}</annotation>

</semantics>

</math></span><img src="./2759b7e26c5b7f286c6ecadec6dbfa550a5400f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.61ex; height:7.176ex;" alt="{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}" loading="lazy"></span></dd></dl>

<p>In this case the change is not <a href="Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>, because every value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> has two corresponding values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}</annotation>

</semantics>

</math></span><img src="./dce25b1a11b8d6f6d6b09a2108360f6ca30a8054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.853ex; height:7.176ex;" alt="{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}" loading="lazy"></span></dd></dl>

<p>The inverse transformation is

</p>

</mrow>

</mrow>

</msubsup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}</annotation>

</semantics>

</math></span><img src="./2220c9f61ec89ce71a29d3980d1f85f7b7c2c6fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.842ex; height:3.676ex;" alt="{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}" loading="lazy"></span></dd></dl>

<p>and its derivative is

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}</annotation>

</semantics>

</math></span><img src="./b166f940d2cdef6c6afda6a17cb2d39c3e646829.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.111ex; height:7.343ex;" alt="{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}" loading="lazy"></span></dd></dl>

<p>Then,

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}</annotation>

</semantics>

</math></span><img src="./be562627f8e548d04f7a3c9569602926d32c4ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.718ex; height:6.176ex;" alt="{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}" loading="lazy"></span></dd></dl>

<p>This is a <a href="Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral chi-squared distribution</a> with one <a href="Degree_of_freedom_(statistics)" class="mw-redirect" title="Degree of freedom (statistics)">degree of freedom</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Some_properties">Some properties</h2></div>

<ul><li>The probability distribution of the sum of two independent random variables is the <b><a href="Convolution" title="Convolution">convolution</a></b> of each of their distributions.</li>

<li>Probability distributions are not a <a href="Vector_space" title="Vector space">vector space</a>—they are not closed under <a href="Linear_combination" title="Linear combination">linear combinations</a>, as these do not preserve non-negativity or total integral 1—but they are closed under <a href="Convex_combination" title="Convex combination">convex combination</a>, thus forming a <a href="Convex_subset" class="mw-redirect" title="Convex subset">convex subset</a> of the space of functions (or measures).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Equivalence_of_random_variables">Equivalence of random variables</h2></div>

<p>There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.

</p><p>In increasing order of strength, the precise definition of these notions of equivalence is given below.

</p>

<div class="mw-heading mw-heading3"><h3 id="Equality_in_distribution">Equality in distribution</h3></div>

<p>If the sample space is a subset of the real line, random variables <i>X</i> and <i>Y</i> are <i>equal in distribution</i> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X{\stackrel {d}{=}}Y}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X{\stackrel {d}{=}}Y}</annotation>

</semantics>

</math></span><img src="./635b086067caf1fb87db7d6670a258111d17b09d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.561ex; height:3.343ex;" alt="{\displaystyle X{\stackrel {d}{=}}Y}" loading="lazy"></span>) if they have the same distribution functions:

</p>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}</annotation>

</semantics>

</math></span><img src="./1ae0a86fb81da98759d3f4a3887695cf223d644c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.195ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}" loading="lazy"></span></dd></dl>

<p>To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal <a href="Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating functions</a> have the same distribution. This provides, for example, a useful method of checking equality of certain functions of <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent, identically distributed (IID) random variables</a>. However, the moment generating function exists only for distributions that have a defined <a href="Laplace_transform" title="Laplace transform">Laplace transform</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Almost_sure_equality">Almost sure equality</h3></div>

<p>Two random variables <i>X</i> and <i>Y</i> are <i>equal <a href="Almost_surely" title="Almost surely">almost surely</a></i> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}</annotation>

</semantics>

</math></span><img src="./2afcd6b106fb5426cd247d7939c13f3281dda5c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.428ex; height:2.843ex;" alt="{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}" loading="lazy"></span>) if, and only if, the probability that they are different is <a href="Null_set" title="Null set">zero</a>:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X\neq Y)=0.}</annotation>

</semantics>

</math></span><img src="./088214863f7b3854c431d827c03f9fd55e017225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.152ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (X\neq Y)=0.}" loading="lazy"></span></dd></dl>

<p>For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

</p>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}</annotation>

</semantics>

</math></span><img src="./b1a0e3e71662425edd3d71d99c9fb274b2ebdf78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.964ex; height:4.343ex;" alt="{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}" loading="lazy"></span></dd></dl>

<p>where "ess sup" represents the <a href="Essential_supremum" class="mw-redirect" title="Essential supremum">essential supremum</a> in the sense of <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Equality">Equality</h3></div>

<p>Finally, the two random variables <i>X</i> and <i>Y</i> are <i>equal</i> if they are equal as functions on their measurable space:

</p>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .}</annotation>

</semantics>

</math></span><img src="./d39fcd62484d01932de2dc0ea922954927c51c78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.503ex; height:2.843ex;" alt="{\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .}" loading="lazy"></span></dd></dl>

<p>This notion is typically the least useful in probability theory because in practice and in theory, the underlying <a href="Measure_space" title="Measure space">measure space</a> of the <a href="Experiment_(probability_theory)" title="Experiment (probability theory)">experiment</a> is rarely explicitly characterized or even characterizable.

</p>

<div class="mw-heading mw-heading3"><h3 id="Practical_difference_between_notions_of_equivalence">Practical difference between notions of equivalence</h3></div>

<p>Since we rarely explicitly construct the probability space underlying a random variable, the difference between these notions of equivalence is somewhat subtle. Essentially, two random variables considered <i>in isolation</i> are "practically equivalent" if they are equal in distribution -- but once we relate them to <i>other</i> random variables defined on the same probability space, then they only remain "practically equivalent" if they are equal almost surely.

</p><p>For example, consider the real random variables <i>A</i>, <i>B</i>, <i>C</i>, and <i>D</i> all defined on the same probability space. Suppose that <i>A</i> and <i>B</i> are equal almost surely (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\;{\stackrel {\text{a.s.}}{=}}\;B}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\;{\stackrel {\text{a.s.}}{=}}\;B}</annotation>

</semantics>

</math></span><img src="./e756565a9c5630bd1797f5ad27fca746d87b0f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.182ex; height:2.843ex;" alt="{\displaystyle A\;{\stackrel {\text{a.s.}}{=}}\;B}" loading="lazy"></span>), but <i>A</i> and <i>C</i> are only equal in distribution (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\stackrel {d}{=}}C}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A{\stackrel {d}{=}}C}</annotation>

</semantics>

</math></span><img src="./366a8e38163149bf4f5ea3977dea13b57d4dac85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.318ex; height:3.343ex;" alt="{\displaystyle A{\stackrel {d}{=}}C}" loading="lazy"></span>). Then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+D\;{\stackrel {\text{a.s.}}{=}}\;B+D}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A+D\;{\stackrel {\text{a.s.}}{=}}\;B+D}</annotation>

</semantics>

</math></span><img src="./8408831419846c62d57ac844bf257ad457045ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.711ex; height:3.009ex;" alt="{\displaystyle A+D\;{\stackrel {\text{a.s.}}{=}}\;B+D}" loading="lazy"></span>, but in general <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+D\;\neq \;C+D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A+D\;\neq \;C+D}</annotation>

</semantics>

</math></span><img src="./e0d66acb288acfdefdca8243f8ca1d6fd759f9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.428ex; height:2.676ex;" alt="{\displaystyle A+D\;\neq \;C+D}" loading="lazy"></span> (not even in distribution). Similarly, we have that the expectation values <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} (AD)=\mathbb {E} (BD)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (AD)=\mathbb {E} (BD)}</annotation>

</semantics>

</math></span><img src="./3c2ca214b2c6395a7e8e86bd4daea5593abb79cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.173ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} (AD)=\mathbb {E} (BD)}" loading="lazy"></span>, but in general <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} (AD)\neq \mathbb {E} (CD)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (AD)\neq \mathbb {E} (CD)}</annotation>

</semantics>

</math></span><img src="./b168410c8fece2b977d8fb3b591c557737bff44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.176ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} (AD)\neq \mathbb {E} (CD)}" loading="lazy"></span>. Therefore, two random variables that are equal in distribution (but not equal almost surely) can have different <a href="Covariance" title="Covariance">covariances</a> with a third random variable.

</p>

<div class="mw-heading mw-heading2"><h2 id="Convergence">Convergence</h2></div>

/* start https://en.wikipedia.org/ */

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></div>

<p>A significant theme in mathematical statistics consists of obtaining convergence results for certain <a href="Sequence" title="Sequence">sequences</a> of random variables; for instance the <a href="Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> and the <a href="Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.

</p><p>There are various senses in which a sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation>

</semantics>

</math></span><img src="./72a8564cedc659cf2f95ae68bc5de2f5207a3285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}" loading="lazy"></span> of random variables can converge to a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. These are explained in the article on <a href="Convergence_of_random_variables" title="Convergence of random variables">convergence of random variables</a>.

</p>

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</style><div class="div-col" style="column-width: 22em;">

<ul><li><a href="Aleatoricism" title="Aleatoricism">Aleatoricism</a></li>

<li><a href="Algebra_of_random_variables" title="Algebra of random variables">Algebra of random variables</a></li>

<li><a href="Event_(probability_theory)" title="Event (probability theory)">Event (probability theory)</a></li>

<li><a href="Multivariate_random_variable" title="Multivariate random variable">Multivariate random variable</a></li>

<li><a href="Pairwise_independence" title="Pairwise independence">Pairwise independent random variables</a></li>

<li><a href="Observable_variable" class="mw-redirect" title="Observable variable">Observable variable</a></li>

<li><a href="Random_compact_set" title="Random compact set">Random compact set</a></li>

<li><a href="Random_element" title="Random element">Random element</a></li>

<li><a href="Random_function" class="mw-redirect" title="Random function">Random function</a></li>

<li><a href="Random_measure" title="Random measure">Random measure</a></li>

<li><a href="Random_number_generator" class="mw-redirect" title="Random number generator">Random number generator</a></li>

<li><a href="Random_variate" title="Random variate">Random variate</a></li>

<li><a href="Random_vector" class="mw-redirect" title="Random vector">Random vector</a></li>

<li><a href="Randomness" title="Randomness">Randomness</a></li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic process</a></li>

<li><a href="Relationships_among_probability_distributions" title="Relationships among probability distributions">Relationships among probability distributions</a></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Inline_citations">Inline citations</h3></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFBlitzsteinHwang2014" class="citation book cs1">Blitzstein, Joe; Hwang, Jessica (2014). <i>Introduction to Probability</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781466575592</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDeisenroth2020" class="citation book cs1">Deisenroth, Marc Peter (2020). <i>Mathematics for machine learning</i>. A. Aldo Faisal, Cheng Soon Ong. Cambridge, United Kingdom: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-108-47004-9</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1104219401">1104219401</a>.</cite></span>

</li>

<li id="cite_note-:3-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-:3_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGeorge_Mackey1980" class="citation journal cs1">George Mackey (July 1980). "Harmonic analysis as the exploitation of symmetry – a historical survey". <i>Bulletin of the American Mathematical Society</i>. New Series. <b>3</b> (1).</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/data/random-variables.html">"Random Variables"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-21</span></span>.</cite></span>

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<li id="cite_note-Yates-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yates_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFYatesMooreStarnes2003" class="citation book cs1">Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/"><i>The Practice of Statistics</i></a> (2nd ed.). New York: <a href="W._H._Freeman_and_Company" title="W. H. Freeman and Company">Freeman</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7167-4773-4</bdi>. Archived from <a rel="nofollow" class="external text" href="http://bcs.whfreeman.com/yates2e/">the original</a> on 2005-02-09.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm">"Random Variables"</a>. <i>www.stat.yale.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-21</span></span>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFDekkingKraaikampLopuhaäMeester2005" class="citation journal cs1">Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/1-84628-168-7">"A Modern Introduction to Probability and Statistics"</a></span>. <i>Springer Texts in Statistics</i>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F1-84628-168-7">10.1007/1-84628-168-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-85233-896-1</bdi>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1431-875X">1431-875X</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFL._CastañedaV._ArunachalamS._Dharmaraja2012" class="citation book cs1">L. Castañeda; V. Arunachalam & S. Dharmaraja (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67"><i>Introduction to Probability and Stochastic Processes with Applications</i></a>. Wiley. p. 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118344941</bdi>.</cite></span>

</li>

<li id="cite_note-:Billingsley-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-:Billingsley_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBillingsley1995" class="citation book cs1">Billingsley, Patrick (1995). <i>Probability and Measure</i> (3rd ed.). Wiley. p. 187. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781466575592</bdi>.</cite></span>

</li>

<li id="cite_note-:0-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_10-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_10-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBertsekas2002" class="citation book cs1">Bertsekas, Dimitri P. (2002). <i>Introduction to Probability</i>. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>188652940X</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/51441829">51441829</a>.</cite></span>

</li>

<li id="cite_note-UCSB-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-UCSB_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSteigerwald" class="citation web cs1">Steigerwald, Douglas G. <a rel="nofollow" class="external text" href="http://faculty.econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf">"Economics 245A – Introduction to Measure Theory"</a> <span class="cs1-format">(PDF)</span>. University of California, Santa Barbara<span class="reference-accessdate">. Retrieved <span class="nowrap">April 26,</span> 2013</span>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFFristedtGray1996">Fristedt & Gray (1996</a>, page 11)</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Literature">Literature</h3></div>

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<ul><li><cite id="CITEREFFristedtGray1996" class="citation book cs1">Fristedt, Bert; Gray, Lawrence (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5D5O8xyM-kMC"><i>A modern approach to probability theory</i></a>. Boston: Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7643-3807-5</bdi>.</cite></li>

<li><cite id="CITEREFBillingsley1995" class="citation book cs1">Billingsley, Patrick (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QyXqOXyxEeIC"><i>Probability and Measure</i></a>. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>8126517719</bdi>.</cite></li>

<li><cite id="CITEREFKallenberg1986" class="citation book cs1"><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bBnvAAAAMAAJ"><i>Random Measures</i></a> (4th ed.). Berlin: <a href="Akademie_Verlag" title="Akademie Verlag">Akademie Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-394960-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0854102">0854102</a>.</cite></li>

<li><cite id="CITEREFKallenberg2001" class="citation book cs1">Kallenberg, Olav (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L6fhXh13OyMC"><i>Foundations of Modern Probability</i></a> (2nd ed.). Berlin: <a href="Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95313-2</bdi>.</cite></li>

<li><cite id="CITEREFPapoulis1965" class="citation book cs1"><a href="Athanasios_Papoulis" title="Athanasios Papoulis">Papoulis, Athanasios</a> (1965). <a rel="nofollow" class="external text" href="http://www.mhhe.com/engcs/electrical/papoulis/"><i>Probability, Random Variables, and Stochastic Processes</i></a> (9th ed.). Tokyo: <a href="McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-119981-0</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Random_variable">"Random variable"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFZukerman2014" class="citation cs2">Zukerman, Moshe (2014), <a rel="nofollow" class="external text" href="http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf"><i>Introduction to Queueing Theory and Stochastic Teletraffic Models</i></a> <span class="cs1-format">(PDF)</span>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1307.2968">1307.2968</a></span></cite></li>

<li><cite id="CITEREFZukerman2014" class="citation cs2">Zukerman, Moshe (2014), <a rel="nofollow" class="external text" href="http://www.ee.cityu.edu.hk/~zukerman/probability.pdf"><i>Basic Probability Topics</i></a> <span class="cs1-format">(PDF)</span></cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="System_identification" title="System identification">System identification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>

<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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