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Cumulative distribution function

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In <a href="Probability_theory" title="Probability theory">probability theory</a> and <a href="Statistics" title="Statistics">statistics</a>, the cumulative distribution function (CDF) of a real-valued <a href="Random_variable" title="Random variable">random variable</a> <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><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">, or just distribution function of <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><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">, evaluated at <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><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">, is the <a href="Probability" title="Probability">probability</a> that <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><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"> will take a value less than or equal to <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><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">.<a href="#cite_note-1">[1]</a>

Every <a href="Probability_distribution" title="Probability distribution">probability distribution</a> <a href="Support_(measure_theory)" title="Support (measure theory)">supported</a> on the real numbers, discrete or "mixed" as well as <a href="Continuous_variable" class="mw-redirect" title="Continuous variable">continuous</a>, is uniquely identified by a <a href="Right-continuous" class="mw-redirect" title="Right-continuous">right-continuous</a> <a href="Monotonic_function" title="Monotonic function">monotone increasing</a> function (a <a href="C%C3%A0dl%C3%A0g" title="Càdlàg">càdlàg</a> function) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon \mathbb {R} \rightarrow [0,1]}">

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

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</math><img src="./b4c45b6faf38bb3fb300ab4678d3675afd172f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.719ex; height:2.843ex;" alt="{\displaystyle F\colon \mathbb {R} \rightarrow [0,1]}" loading="lazy"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}">

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}</annotation>

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</math><img src="./d5624f61bba2a30c533cc18a8e50b11c65313590.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.033ex; height:4.009ex;" alt="{\displaystyle \lim _{x\rightarrow -\infty }F(x)=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\rightarrow \infty }F(x)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow \infty }F(x)=1}</annotation>

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</math><img src="./1653d3d823bf499f00c6de578df43b16da3db35f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.754ex; height:3.843ex;" alt="{\displaystyle \lim _{x\rightarrow \infty }F(x)=1}" loading="lazy">.

In the case of a scalar <a href="Continuous_distribution" class="mw-redirect" title="Continuous distribution">continuous distribution</a>, it gives the area under the <a href="Probability_density_function" title="Probability density function">probability density function</a> from negative infinity to <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><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">. Cumulative distribution functions are also used to specify the distribution of <a href="Multivariate_random_variable" title="Multivariate random variable">multivariate random variables</a>.

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

The cumulative distribution function of a real-valued <a href="Random_variable" title="Random variable">random variable</a> <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><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"> is the function given by<a href="#cite_note-KunIlPark-2">[2]</a>: 77

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

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</math><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">   (Eq.1)

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where the right-hand side represents the <a href="Probability" title="Probability">probability</a> that the random variable <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><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"> takes on a value less than or equal to <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><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">.

The probability that <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><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"> lies in the semi-closed <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b]}</annotation>

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</math><img src="./6a6969e731af335df071e247ee7fb331cd1a57ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="{\displaystyle (a,b]}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}">

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

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</math><img src="./91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy">, is therefore<a href="#cite_note-KunIlPark-2">[2]</a>: 84

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

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</math><img src="./2cd46015ef99275b537dd0d116b00721a216b2cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.835ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (a<X\leq b)=F_{X}(b)-F_{X}(a)}" loading="lazy">   (Eq.2)

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In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the <a href="Binomial_distribution" title="Binomial distribution">binomial</a> and <a href="Poisson_distribution" title="Poisson distribution">Poisson distributions</a> depends upon this convention. Moreover, important formulas like <a href="Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a>'s inversion formula for the <a href="Characteristic_function_(probability_theory)#Inversion_formula" title="Characteristic function (probability theory)">characteristic function</a> also rely on the "less than or equal" formulation.

If treating several random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y,\ldots }">

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

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</math><img src="./d0677d20d4752776a420872ba304c72902a82716.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:2.509ex;" alt="{\displaystyle X,Y,\ldots }" loading="lazy"> etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><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"> for a cumulative distribution function, in contrast to the lower-case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> used for <a href="Probability_density_function" title="Probability density function">probability density functions</a> and <a href="Probability_mass_function" title="Probability mass function">probability mass functions</a>. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the <a href="Normal_distribution" title="Normal distribution">normal distribution</a> uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, respectively.

The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating<a href="#cite_note-3">[3]</a> using the <a href="Fundamental_Theorem_of_Calculus" class="mw-redirect" title="Fundamental Theorem of Calculus">Fundamental Theorem of Calculus</a>; i.e. given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

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

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</math><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">,

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

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as long as the derivative exists.

The CDF of a <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a> <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><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"> can be expressed as the integral of its probability density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}}">

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

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</math><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"> as follows:<a href="#cite_note-KunIlPark-2">[2]</a>: 86

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<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt.}</annotation>

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In the case of a random variable <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><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"> which has distribution having a discrete component at a value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x).}</annotation>

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If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}}">

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

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</math><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">, this equals zero and there is no discrete component at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">.

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

Every cumulative distribution function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}}">

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

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</math><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> is <a href="Monotone_increasing" class="mw-redirect" title="Monotone increasing">non-decreasing</a><a href="#cite_note-KunIlPark-2">[2]</a>: 78 and <a href="Right-continuous" class="mw-redirect" title="Right-continuous">right-continuous</a>,<a href="#cite_note-KunIlPark-2">[2]</a>: 79 which makes it a <a href="C%C3%A0dl%C3%A0g" title="Càdlàg">càdlàg</a> function. Furthermore,

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</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }F_{X}(x)=0,\quad \lim _{x\to +\infty }F_{X}(x)=1.}</annotation>

</semantics>

</math>

Every function with these three properties is a CDF, i.e., for every such function, a <a href="Random_variable" title="Random variable">random variable</a> can be defined such that the function is the cumulative distribution function of that random variable.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is a purely <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a>, then it attains values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\ldots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3bfc8b10a5ede4dae0d4341edcc65f6a20232e00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.559ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\ldots }" loading="lazy"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}=p(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3ed11157e8358455d575ed0c10215b7b39d7d1db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.265ex; height:2.843ex;" alt="{\displaystyle p_{i}=p(x_{i})}" loading="lazy">, and the CDF of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> will be <a href="Discontinuity_(mathematics)" class="mw-redirect" title="Discontinuity (mathematics)">discontinuous</a> at the points <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><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">:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i}).}</annotation>

</semantics>

</math>

If the CDF <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><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> of a real valued random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is <a href="Continuous_function" title="Continuous function">continuous</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is a <a href="Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a>; if furthermore <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><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> is <a href="Absolute_continuity" title="Absolute continuity">absolutely continuous</a>, then there exists a <a href="Lebesgue_integral" title="Lebesgue integral">Lebesgue-integrable</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./29310a010e7f7dfa33ba69bcf1ef9ec166d461dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)}" loading="lazy"> such that

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(b)-F_{X}(a)=\operatorname {P} (a<X\leq b)=\int _{a}^{b}f_{X}(x)\,dx}</annotation>

</semantics>

</math>

for all real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">. The function <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><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"> is equal to the <a href="Derivative" title="Derivative">derivative</a> of <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><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>, and it is called the <a href="Probability_density_function" title="Probability density function">probability density function</a> of the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> has finite <a href="L1-norm" class="mw-redirect" title="L1-norm">L1-norm</a>, that is, the expectation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffa803ac0386dbd3c04054a71776b4edc435151c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.274ex; height:2.843ex;" alt="{\displaystyle |X|}" loading="lazy"> is finite, then the expectation is given by the <a href="Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X]=\int _{-\infty }^{\infty }t\,dF_{X}(t)}</annotation>

</semantics>

</math>

and for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy">,

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(1-F_{X}(x))\leq \int _{x}^{\infty }t\,dF_{X}(t)}</annotation>

</semantics>

</math>

as well as

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xF_{X}(-x)\leq \int _{-\infty }^{-x}(-t)\,dF_{X}(t)}</annotation>

</semantics>

</math>

as shown in the diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of <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><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy">). In particular, we have

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to -\infty }xF_{X}(x)=0,\quad \lim _{x\to +\infty }x(1-F_{X}(x))=0.}</annotation>

</semantics>

</math>

In addition, the (finite) expected value of the real-valued random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> can be defined on the graph of its cumulative distribution function as illustrated by the <a href="Expected_value#Uhl2023Bild1" title="Expected value">drawing</a> in the <a href="Expected_value#Arbitrary_real-valued_random_variables" title="Expected value">definition of expected value for arbitrary real-valued random variables</a>.

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

As an example, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> on the unit interval <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><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">.

Then the CDF of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is given by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ x>1\end{cases}}}">

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ x>1\end{cases}}}</annotation>

</semantics>

</math>

Suppose instead that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> takes only the discrete values 0 and 1, with equal probability.

Then the CDF of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is given by

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\1/2&:\ 0\leq x<1\\1&:\ x\geq 1\end{cases}}}</annotation>

</semantics>

</math>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is <a href="Exponential_distribution" title="Exponential distribution">exponential distributed</a>. Then the CDF of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is given by

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}</annotation>

</semantics>

</math>

Here λ > 0 is the parameter of the distribution, often called the rate parameter.

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is <a href="Normal_distribution" title="Normal distribution">normal distributed</a>. Then the CDF of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is given by

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(t;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{t}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,dx.}</annotation>

</semantics>

</math>

Here the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is the <a href="Mean" title="Mean">mean</a> or expectation of the distribution; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"> is its standard deviation.

A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the <a href="Standard_normal_table" title="Standard normal table">standard normal table</a>, the unit normal table, or the Z table.

Suppose <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><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"> is <a href="Binomial_distribution" title="Binomial distribution">binomial distributed</a>. Then the CDF of <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><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"> is given by

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<annotation encoding="application/x-tex">{\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i}}</annotation>

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</math>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> independent experiments, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor k\rfloor }">

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

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</math><img src="./0a64054ec38bec048113dd328cf867e9e62b581d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.276ex; height:2.843ex;" alt="{\displaystyle \lfloor k\rfloor }" loading="lazy"> is the "floor" under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, i.e. the greatest integer less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Derived_functions">Derived functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Complementary_cumulative_distribution_function_(tail_distribution)">Complementary cumulative distribution function (tail distribution)</h3></div>

Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as

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<annotation encoding="application/x-tex">{\displaystyle {\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).}</annotation>

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</math>

This has applications in <a href="Statistics" title="Statistics">statistical</a> <a href="Hypothesis_test" class="mw-redirect" title="Hypothesis test">hypothesis testing</a>, for example, because the one-sided <a href="P-value" title="P-value">p-value</a> is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the <a href="Test_statistic" title="Test statistic">test statistic</a>, T, has a continuous distribution, the one-sided <a href="P-value" title="P-value">p-value</a> is simply given by the ccdf: for an observed value <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><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> of the test statistic

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<annotation encoding="application/x-tex">{\displaystyle p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).}</annotation>

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</math>

In <a href="Survival_analysis" title="Survival analysis">survival analysis</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {F}}_{X}(x)}">

<msub>

<mover>

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

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</math><img src="./e49da3eba602170ffcff8e8d4e6d1319ef2d83c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.576ex; height:3.009ex;" alt="{\displaystyle {\bar {F}}_{X}(x)}" loading="lazy"> is called the <a href="Survival_function" title="Survival function">survival function</a> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)}">

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

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</math><img src="./1886b5a535ed8f168a7c3a83afc8ca440edcdc6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle S(x)}" loading="lazy">, while the term reliability function is common in <a href="Engineering" title="Engineering">engineering</a>.

<dl><dt>Properties</dt></dl>

<ul><li>For a non-negative continuous random variable having an expectation, <a href="Markov's_inequality" title="Markov's inequality">Markov's inequality</a> states that<a href="#cite_note-ZK-4">[4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {F}}_{X}(x)\leq {\frac {\operatorname {E} (X)}{x}}.}">

<msub>

<mover>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {F}}_{X}(x)\leq {\frac {\operatorname {E} (X)}{x}}.}</annotation>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle x\to \infty ,{\bar {F}}_{X}(x)\to 0}</annotation>

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</math><img src="./5589ad87b92c43377d4d5155cbbc70dc26dfa508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.654ex; height:3.009ex;" alt="{\displaystyle x\to \infty ,{\bar {F}}_{X}(x)\to 0}" loading="lazy">, and in fact <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {F}}_{X}(x)=o(1/x)}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {F}}_{X}(x)=o(1/x)}</annotation>

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</math><img src="./d3c36609acb96968736d887e36589515f9547afd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.266ex; height:3.009ex;" alt="{\displaystyle {\bar {F}}_{X}(x)=o(1/x)}" loading="lazy"> provided that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)}">

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

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</math><img src="./734c66846ec383f7e00c23ae4f4155f3cdc8c0fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (X)}" loading="lazy"> is finite. Proof: Assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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

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</math><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"> has a density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}}">

<msub>

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

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</math><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">, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

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

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</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }xf_{X}(x)\,dx\geq \int _{0}^{c}xf_{X}(x)\,dx+c\int _{c}^{\infty }f_{X}(x)\,dx}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }xf_{X}(x)\,dx\geq \int _{0}^{c}xf_{X}(x)\,dx+c\int _{c}^{\infty }f_{X}(x)\,dx}</annotation>

</semantics>

</math> Then, on recognizing <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {F}}_{X}(c)=\int _{c}^{\infty }f_{X}(x)\,dx}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {F}}_{X}(c)=\int _{c}^{\infty }f_{X}(x)\,dx}</annotation>

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</math> and rearranging terms, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq c{\bar {F}}_{X}(c)\leq \operatorname {E} (X)-\int _{0}^{c}xf_{X}(x)\,dx\to 0{\text{ as }}c\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq c{\bar {F}}_{X}(c)\leq \operatorname {E} (X)-\int _{0}^{c}xf_{X}(x)\,dx\to 0{\text{ as }}c\to \infty }</annotation>

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</math> as claimed.</li>

<li>For a random variable having an expectation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }{\bar {F}}_{X}(x)\,dx-\int _{-\infty }^{0}F_{X}(x)\,dx}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }{\bar {F}}_{X}(x)\,dx-\int _{-\infty }^{0}F_{X}(x)\,dx}</annotation>

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</math> and for a non-negative random variable the second term is 0. If the random variable can only take non-negative integer values, this is equivalent to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)=\sum _{n=0}^{\infty }{\bar {F}}_{X}(n).}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=\sum _{n=0}^{\infty }{\bar {F}}_{X}(n).}</annotation>

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</math></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Folded_cumulative_distribution">Folded cumulative distribution</h3></div>

While the plot of a cumulative distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><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"> often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over,<a href="#cite_note-Gentle-5">[5]</a><a href="#cite_note-Monti-6">[6]</a> that is

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<annotation encoding="application/x-tex">{\displaystyle F_{\text{fold}}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}}</annotation>

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</math><img src="./64fb171fc50495773d22a592a91603af18b6170b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:48.811ex; height:3.176ex;" alt="{\displaystyle F_{\text{fold}}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\{A\}}}">

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

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</math><img src="./51d9d2bbebabb1349814da07d47f898237b12c58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.271ex; height:3.009ex;" alt="{\displaystyle 1_{\{A\}}}" loading="lazy"> denotes the <a href="Indicator_function" title="Indicator function">indicator function</a> and the second summand is the <a href="Survivor_function" class="mw-redirect" title="Survivor function">survivor function</a>, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the <a href="Median_(statistics)" class="mw-redirect" title="Median (statistics)">median</a>, <a href="Dispersion_(statistics)" class="mw-redirect" title="Dispersion (statistics)">dispersion</a> (specifically, the <a href="Mean_absolute_deviation" class="mw-redirect" title="Mean absolute deviation">mean absolute deviation</a> from the median<a href="#cite_note-7">[7]</a>) and <a href="Skewness" title="Skewness">skewness</a> of the distribution or of the empirical results.

<div class="mw-heading mw-heading3"><h3 id="Inverse_distribution_function_(quantile_function)">Inverse distribution function (quantile function)</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Quantile_function" title="Quantile function">Quantile function</a></div>

If the CDF F is strictly increasing and continuous then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(p),p\in [0,1],}">

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<annotation encoding="application/x-tex">{\displaystyle F^{-1}(p),p\in [0,1],}</annotation>

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</math><img src="./b89fe1b58ff06ad5647ba178886bb6704d8da846.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.47ex; height:3.176ex;" alt="{\displaystyle F^{-1}(p),p\in [0,1],}" loading="lazy"> is the unique real number <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><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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=p}">

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

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</math><img src="./e5419ccdd19e08d660f88a3915e14f2ee6535a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.148ex; height:2.843ex;" alt="{\displaystyle F(x)=p}" loading="lazy">. This defines the inverse distribution function or <a href="Quantile_function" title="Quantile function">quantile function</a>.

Some distributions do not have a unique inverse (for example if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)=0}">

<msub>

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

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</math><img src="./d7764ffd6e384963d487c2dfd4a5a183c75f3ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.171ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)=0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<x<b}">

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

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</math><img src="./e09d6f5cf7cd5aab0408e75dae5f10109ae9fce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.754ex; height:2.176ex;" alt="{\displaystyle a<x<b}" loading="lazy">, causing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}}">

<msub>

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

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</math><img src="./062f285db773e329f6c270cb6b65fa076996c941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle F_{X}}" loading="lazy"> to be constant). In this case, one may use the generalized inverse distribution function, which is defined as

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<annotation encoding="application/x-tex">{\displaystyle F^{-1}(p)=\inf\{x\in \mathbb {R} :F(x)\geq p\},\quad \forall p\in [0,1].}</annotation>

</semantics>

</math><img src="./d7f8d361fa630d92216975a16f12756a137f3d84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.093ex; height:3.176ex;" alt="{\displaystyle F^{-1}(p)=\inf\{x\in \mathbb {R} :F(x)\geq p\},\quad \forall p\in [0,1].}" loading="lazy"></dd></dl>

<ul><li>Example 1: The median is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(0.5)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{-1}(0.5)}</annotation>

</semantics>

</math><img src="./cae637f92358e15bbf6c469e49934f04733b5561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.929ex; height:3.176ex;" alt="{\displaystyle F^{-1}(0.5)}" loading="lazy">.</li>

<li>Example 2: Put <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =F^{-1}(0.95)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =F^{-1}(0.95)}</annotation>

</semantics>

</math><img src="./ab15baf971d9ed12e81a4842b23ab574d37ab6ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.391ex; height:3.176ex;" alt="{\displaystyle \tau =F^{-1}(0.95)}" loading="lazy">. Then we call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> the 95th percentile.</li></ul>

Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized <a href="Inverse_distribution" title="Inverse distribution">inverse distribution</a> function) are:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e2a1a300038291319dc37f8cf5d1e87b6abc1ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.148ex; height:2.676ex;" alt="{\displaystyle F^{-1}}" loading="lazy"> is nondecreasing<a href="#cite_note-8">[8]</a></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2a5d210f02caede50cbf64cbf860c5f71921095c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.265ex; height:3.176ex;" alt="{\displaystyle F^{-1}(F(x))\leq x}" loading="lazy"></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(F^{-1}(p))\geq p}</annotation>

</semantics>

</math><img src="./d1e6a4a1077ea65bc627315a1f756b4ddac34e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.944ex; height:3.176ex;" alt="{\displaystyle F(F^{-1}(p))\geq p}" loading="lazy"></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{-1}(p)\leq x}</annotation>

</semantics>

</math><img src="./ba6e0abf615b6674741dadf13bdf108ae4f25f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.554ex; height:3.176ex;" alt="{\displaystyle F^{-1}(p)\leq x}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq F(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./61b9ad1c0c3e109c8af6686d7defed3cd7fc5d65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.237ex; height:2.843ex;" alt="{\displaystyle p\leq F(x)}" loading="lazy"></li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> has a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U[0,1]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ef4a82b2a883d751cf53e5ac11ea12b9e36298f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.435ex; height:2.843ex;" alt="{\displaystyle U[0,1]}" loading="lazy"> distribution then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(Y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f29332580f17f816e0aa118594c752979593e546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.73ex; height:3.176ex;" alt="{\displaystyle F^{-1}(Y)}" loading="lazy"> is distributed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. This is used in <a href="Random_number_generation" title="Random number generation">random number generation</a> using the <a href="Inverse_transform_sampling" title="Inverse transform sampling">inverse transform sampling</a>-method.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6e6cf67666024497eeac814dcbd887dca34288a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.533ex; height:2.843ex;" alt="{\displaystyle \{X_{\alpha }\}}" loading="lazy"> is a collection of independent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-distributed random variables defined on the same <a href="Sample_space" title="Sample space">sample space</a>, then there exist random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3c46f766b7763aac14d4c6fd969a90d955cc6e63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.635ex; height:2.509ex;" alt="{\displaystyle Y_{\alpha }}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3c46f766b7763aac14d4c6fd969a90d955cc6e63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.635ex; height:2.509ex;" alt="{\displaystyle Y_{\alpha }}" loading="lazy"> is distributed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U[0,1]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ef4a82b2a883d751cf53e5ac11ea12b9e36298f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.435ex; height:2.843ex;" alt="{\displaystyle U[0,1]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(Y_{\alpha })=X_{\alpha }}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{-1}(Y_{\alpha })=X_{\alpha }}</annotation>

</semantics>

</math><img src="./99979a4d4b66f3cad51fa37e7f974deedf214151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.898ex; height:3.176ex;" alt="{\displaystyle F^{-1}(Y_{\alpha })=X_{\alpha }}" loading="lazy"> with probability 1 for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">.</li></ol>

The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.

<div class="mw-heading mw-heading3"><h3 id="Empirical_distribution_function">Empirical distribution function</h3></div>

The <a href="Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the <a href="Rate_of_convergence" title="Rate of convergence">rate of convergence</a> of the empirical distribution function to the underlying cumulative distribution function.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Multivariate_case">Multivariate case</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_for_two_random_variables">Definition for two random variables</h3></div>

When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy">, the joint CDF <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{XY}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94c1320397f88d8f9bc162f9cacbc311de8198e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.381ex; height:2.509ex;" alt="{\displaystyle F_{XY}}" loading="lazy"> is given by<a href="#cite_note-KunIlPark-2">[2]</a>: 89

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./78c9e857451ec29be554d6c5d28b7174e18b9d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.127ex; height:3.009ex;" alt="{\displaystyle F_{X,Y}(x,y)=\operatorname {P} (X\leq x,Y\leq y)}" loading="lazy">   (Eq.3)

</div>

where the right-hand side represents the <a href="Probability" title="Probability">probability</a> that the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> takes on a value less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> takes on a value less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.

Example of joint cumulative distribution function:

For two continuous variables X and Y: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;}</annotation>

</semantics>

</math>

For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example:<a href="#cite_note-10">[10]</a>

given the joint probability mass function in tabular form, determine the joint cumulative distribution function.

<td>

</td>

<td>Y = 2

</td>

<td>Y = 4

</td>

<td>Y = 6

</td>

<td>Y = 8

</td></tr>

<tr>

<td>X = 1

</td>

<td>0

</td>

<td>0.1

</td>

<td>0

</td>

<td>0.1

</td></tr>

<tr>

<td>X = 3

</td>

<td>0

</td>

<td>0

</td>

<td>0.2

</td>

<td>0

</td></tr>

<tr>

<td>X = 5

</td>

<td>0.3

</td>

<td>0

</td>

<td>0

</td>

<td>0.15

</td></tr>

<tr>

<td>X = 7

</td>

<td>0

</td>

<td>0

</td>

<td>0.15

</td>

<td>0

</td></tr></tbody></table>

Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form:

<td>

</td>

<td>Y < 2

</td>

<td>Y ≤ 2

</td>

<td>Y ≤ 4

</td>

<td>Y ≤ 6

</td>

<td>Y ≤ 8

</td></tr>

<tr>

<td>X < 1

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td></tr>

<tr>

<td>X ≤ 1

</td>

<td>0

</td>

<td>0

</td>

<td>0.1

</td>

<td>0.1

</td>

<td>0.2

</td></tr>

<tr>

<td>X ≤ 3

</td>

<td>0

</td>

<td>0

</td>

<td>0.1

</td>

<td>0.3

</td>

<td>0.4

</td></tr>

<tr>

<td>X ≤ 5

</td>

<td>0

</td>

<td>0.3

</td>

<td>0.4

</td>

<td>0.6

</td>

<td>0.85

</td></tr>

<tr>

<td>X ≤ 7

</td>

<td>0

</td>

<td>0.3

</td>

<td>0.4

</td>

<td>0.75

</td>

<td>1

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Definition_for_more_than_two_random_variables">Definition for more than two random variables</h3></div>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{N}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1ddd30bbaef612ab5c5f7ea6f1a18646cde0c7f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.772ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{N}}" loading="lazy">, the joint CDF <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X_{1},\ldots ,X_{N}}}">

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9174bd581de2144c4cba0d44928d6f1142de87cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.469ex; height:2.843ex;" alt="{\displaystyle F_{X_{1},\ldots ,X_{N}}}" loading="lazy"> is given by

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X_{1},\ldots ,X_{N}}(x_{1},\ldots ,x_{N})=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}</annotation>

</semantics>

</math><img src="./b58f5cd681f18da51246c2c87b2e0223e01d64a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.727ex; height:3.009ex;" alt="{\displaystyle F_{X_{1},\ldots ,X_{N}}(x_{1},\ldots ,x_{N})=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}" loading="lazy">   (Eq.4)

</div>

Interpreting the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> random variables as a <a href="Random_vector" class="mw-redirect" title="Random vector">random vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}}</annotation>

</semantics>

</math><img src="./81fe6f1eb53e42a584ec58900302fe672d6c8082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.089ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}}" loading="lazy"> yields a shorter notation:

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<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

Every multivariate CDF is:

<ol><li>Monotonically non-decreasing for each of its variables,</li>

<li>Right-continuous in each of its variables,</li>

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<annotation encoding="application/x-tex">{\displaystyle 0\leq F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})\leq 1,}</annotation>

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</math><img src="./96d249847f967fbb8b7ebbd8185d1fd85865f620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.259ex; height:3.009ex;" alt="{\displaystyle 0\leq F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})\leq 1,}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x_{1},\ldots ,x_{n}\to +\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=1}</annotation>

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</math><img src="./8901acb65298079e4a0de36f710b168b7f81f782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:35.82ex; height:4.176ex;" alt="{\displaystyle \lim _{x_{1},\ldots ,x_{n}\to +\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x_{i}\to -\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=0,}">

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x_{i}\to -\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=0,}</annotation>

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</math><img src="./2c69c0f43e588e1c035cf2f03f73f8c2c471fd83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.515ex; height:4.176ex;" alt="{\displaystyle \lim _{x_{i}\to -\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=0,}" loading="lazy"> for all i.</li></ol>

Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=0}">

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

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</math><img src="./c910dd64922e23f212bc46e970af0540811c48f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=0}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0}">

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

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</math><img src="./1a4dbbf970b2d2863dcab589eafe006f08e727d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y<1}">

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

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</math><img src="./9e3983799a9e0f1a501911883535824d04710394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x+y<1}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<0}">

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

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</math><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"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)=1}">

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

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</math><img src="./213396503b3522912d07214b683b010ad12ab3b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.33ex; height:2.843ex;" alt="{\displaystyle F(x,y)=1}" loading="lazy"> otherwise. It is easy to see that the above conditions are met, and yet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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

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</math><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"> is not a CDF since if it was, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {P} \left({\frac {1}{3}}<X\leq 1,{\frac {1}{3}}<Y\leq 1\right)=-1}">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {P} \left({\frac {1}{3}}<X\leq 1,{\frac {1}{3}}<Y\leq 1\right)=-1}</annotation>

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</math><img src="./1da9d1ac26d9c7b1d83d2bdfe53d5a2747c2cfa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.25ex; height:4.843ex;" alt="{\textstyle \operatorname {P} \left({\frac {1}{3}}<X\leq 1,{\frac {1}{3}}<Y\leq 1\right)=-1}" loading="lazy"> as explained below.

The probability that a point belongs to a <a href="Hyperrectangle" title="Hyperrectangle">hyperrectangle</a> is analogous to the 1-dimensional case:<a href="#cite_note-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{X_{1},X_{2}}(a,c)+F_{X_{1},X_{2}}(b,d)-F_{X_{1},X_{2}}(a,d)-F_{X_{1},X_{2}}(b,c)=\operatorname {P} (a<X_{1}\leq b,c<X_{2}\leq d)=\int \cdots }</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Complex_case">Complex case</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Complex_random_variable">Complex random variable</h3></div>

The generalization of the cumulative distribution function from real to <a href="Complex_random_variable#Cumulative_distribution_function" title="Complex random variable">complex random variables</a> is not obvious because expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(Z\leq 1+2i)}">

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<annotation encoding="application/x-tex">{\displaystyle P(Z\leq 1+2i)}</annotation>

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</math><img src="./fefbab265075b4193f374e6d3387a2758e683b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.301ex; height:2.843ex;" alt="{\displaystyle P(Z\leq 1+2i)}" loading="lazy"> make no sense. However expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}">

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<annotation encoding="application/x-tex">{\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}</annotation>

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</math><img src="./0c95a79687e41138cf6171b5770d39dcacd47fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.302ex; height:2.843ex;" alt="{\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}" loading="lazy"> make sense. Therefore, we define the cumulative distribution of a complex random variables via the <a href="Joint_probability_distribution" title="Joint probability distribution">joint distribution</a> of their real and imaginary parts:

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<annotation encoding="application/x-tex">{\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)}).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Complex_random_vector">Complex random vector</h3></div>

Generalization of <a href="#math_Eq.4">Eq.4</a> yields

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{\mathbf {Z} }(\mathbf {z} )&=F_{\Re {(Z_{1})},\Im {(Z_{1})},\ldots ,\Re {(Z_{n})},\Im {(Z_{n})}}(\Re {(z_{1})},\Im {(z_{1})},\ldots ,\Re {(z_{n})},\Im {(z_{n})})\\[1ex]&=\operatorname {P} (\Re {(Z_{1})}\leq \Re {(z_{1})},\Im {(Z_{1})}\leq \Im {(z_{1})},\ldots ,\Re {(Z_{n})}\leq \Re {(z_{n})},\Im {(Z_{n})}\leq \Im {(z_{n})})\end{aligned}}}</annotation>

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as definition for the CDS of a complex random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{N})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{N})^{T}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Use_in_statistical_analysis">Use in statistical analysis</h2></div>

The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. <a href="Cumulative_frequency_analysis" title="Cumulative frequency analysis">Cumulative frequency analysis</a> is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The <a href="Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various <a href="Statistical_hypothesis_test" title="Statistical hypothesis test">statistical hypothesis tests</a>. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.

<div class="mw-heading mw-heading3"><h3 id="Kolmogorov–Smirnov_and_Kuiper's_tests">Kolmogorov–Smirnov and Kuiper's tests</h3></div>

The <a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a> is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related <a href="Kuiper's_test" title="Kuiper's test">Kuiper's test</a> is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.

<ul><li><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></li>

<li><a href="Distribution_fitting" class="mw-redirect" title="Distribution fitting">Distribution fitting</a></li>

<li><a href="Ogive_(statistics)" title="Ogive (statistics)">Ogive (statistics)</a></li></ul>

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

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</style><cite id="CITEREFDeisenrothFaisalOng2020" class="citation book cs1">Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon (2020). <a rel="nofollow" class="external text" href="https://github.com/mml-book/mml-book.github.io">Mathematics for Machine Learning</a>. Cambridge University Press. p. 181. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781108455145</bdi>.</cite>

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<li id="cite_note-KunIlPark-2">^ <a href="#cite_ref-KunIlPark_2-0">a</a> <a href="#cite_ref-KunIlPark_2-1">b</a> <a href="#cite_ref-KunIlPark_2-2">c</a> <a href="#cite_ref-KunIlPark_2-3">d</a> <a href="#cite_ref-KunIlPark_2-4">e</a> <a href="#cite_ref-KunIlPark_2-5">f</a> <cite id="CITEREFPark2018" class="citation book cs1">Park, Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-68074-3</bdi>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMontgomeryRunger2003" class="citation book cs1">Montgomery, Douglas C.; Runger, George C. (2003). <a rel="nofollow" class="external text" href="http://www.um.edu.ar/math/montgomery.pdf">Applied Statistics and Probability for Engineers</a> (PDF). John Wiley & Sons, Inc. p. 104. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-20454-4</bdi>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120730233253/http://www.um.edu.ar/math/montgomery.pdf">Archived</a> (PDF) from the original on 2012-07-30.</cite>

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<cite id="CITEREFMonti1995" class="citation journal cs1"><a href="Katherine_Monti" title="Katherine Monti">Monti, K. L.</a> (1995). "Folded Empirical Distribution Function Curves (Mountain Plots)". The American Statistician. 49 (4): 342–345. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2684570">10.2307/2684570</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2684570">2684570</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFXueTitterington2011" class="citation journal cs1">Xue, J. H.; Titterington, D. M. (2011). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-00753950/file/PEER_stage2_10.1016%252Fj.spl.2011.03.014.pdf">"The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile"</a> (PDF). Statistics & Probability Letters. 81 (8): 1179–1182. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spl.2011.03.014">10.1016/j.spl.2011.03.014</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFChan2021" class="citation book cs1">Chan, Stanley H. (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GR2jzgEACAAJ">Introduction to Probability for Data Science</a>. Michigan Publishing. p. 18. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-60785-746-4</bdi>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFHesse1990" class="citation journal cs1">Hesse, C. (1990). "Rates of convergence for the empirical distribution function and the empirical characteristic function of a broad class of linear processes". Journal of Multivariate Analysis. 35 (2): 186–202. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0047-259X%2890%2990024-C">10.1016/0047-259X(90)90024-C</a>.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.info/Probability/Joint_CDF/">"Joint Cumulative Distribution Function (CDF)"</a>. math.info. Retrieved 2019-12-11.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160222051842/http://www.math.wustl.edu/~hgan/Prob2014/slides.259-327.pdf">"Archived copy"</a> (PDF). www.math.wustl.edu. Archived from <a rel="nofollow" class="external text" href="http://www.math.wustl.edu/~hgan/Prob2014/slides.259-327.pdf">the original</a> (PDF) on 22 February 2016. Retrieved 13 January 2022.</cite><code class="cs1-code">{{cite web}}</code>: CS1 maint: archived copy as title (link)

</li>

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

<ul><li> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Cumulative_distribution_functions" class="extiw external" title="commons:Category:Cumulative distribution functions">Cumulative distribution functions</a> at Wikimedia Commons</li></ul>

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</style><div id="Theory_of_probability_distributions39" style="font-size:114%;margin:0 4em">Theory of <a href="Probability_distribution" title="Probability distribution">probability distributions</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Probability_mass_function" title="Probability mass function">probability mass function</a> (pmf)</li>

<li><a href="Probability_density_function" title="Probability density function">probability density function</a> (pdf)</li>

<li><a href="Quantile_function" title="Quantile function">quantile function</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Raw_moment" class="mw-redirect" title="Raw moment">raw moment</a></li>

<li><a href="Central_moment" title="Central moment">central moment</a></li>

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

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

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

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

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

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Moment-generating_function" title="Moment-generating function">moment-generating function</a> (mgf)</li>

<li><a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a></li>

<li><a href="Probability-generating_function" title="Probability-generating function">probability-generating function</a> (pgf)</li>

<li><a href="Cumulant" title="Cumulant">cumulant</a></li>

<li><a href="Combinant" title="Combinant">combinant</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q386228#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata529" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q386228#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata529" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</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 rel="nofollow" class="external text" href="https://d-nb.info/gnd/4192219-0">Germany</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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 rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/37253c20-16a3-419f-876c-bdc7dca04485">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-07" href="https://en.wikipedia.org/wiki/?title=Cumulative_distribution_function&oldid=1304688259">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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