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Probability density function

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In <a href="Probability_theory" title="Probability theory">probability theory</a>, a probability density function (PDF), density function, or density of an <a href="Absolutely_continuous_random_variable" class="mw-redirect" title="Absolutely continuous random variable">absolutely continuous random variable</a>, is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose value at any given sample (or point) in the <a href="Sample_space" title="Sample space">sample space</a> (the set of possible values taken by the random variable) can be interpreted as providing a <a href="Relative_likelihood" title="Relative likelihood">relative likelihood</a> that the value of the random variable would be equal to that sample.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> Probability density is the probability per unit length, in other words. While the absolute likelihood for a continuous <a href="Random_variable" title="Random variable">random variable</a> to take on any particular value is zero, given there is an infinite set of possible values to begin with. Therefore, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the <a href="Random_variable" title="Random variable">random variable</a> falling within a particular range of values, as opposed to taking on any one value. This probability is given by the <a href="Integral" title="Integral">integral</a> of a continuous variable's PDF over that range, where the integral is the nonnegative area under the density function between the lowest and greatest values of the range. The PDF is nonnegative everywhere, and the area under the entire curve is equal to one, such that the probability of the random variable falling within the set of possible values is 100%.

The terms probability distribution function and probability function can also denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> is defined as a function over general sets of values or it may refer to the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF), or it may be a <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> (PMF) rather than the density. Density function itself is also used for the probability mass function, leading to further confusion.<a href="#cite_note-4">[4]</a> In general the PMF is used in the context of <a href="Continuous_or_discrete_variable#Discrete_variable" title="Continuous or discrete variable">discrete random variables</a> (random variables that take values on a <a href="Countable_set" title="Countable set">countable set</a>), while the PDF is used in the context of continuous random variables.

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

Suppose bacteria of a certain species typically live 20 to 30 hours. The probability that a bacterium lives exactly 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.

In this example, the ratio (probability of living during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour−1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as (2 hour−1) dt. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ 6×10−13 (using the <a href="Conversion_of_units" title="Conversion of units">unit conversion</a> 3.6×1012 nanoseconds = 1 hour).

There is a probability density function f with f(5 hours) = 2 hour−1. The <a href="Integral" title="Integral">integral</a> of f over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.

<div class="mw-heading mw-heading2"><h2 id="Absolutely_continuous_univariate_distributions">Absolutely continuous univariate distributions</h2></div>

A probability density function is most commonly associated with <a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">absolutely continuous</a> <a href="Univariate_distribution" title="Univariate distribution">univariate distributions</a>. A <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 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">, where <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"> is a non-negative <a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue-integrable</a> function, if:

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<annotation encoding="application/x-tex">{\displaystyle \Pr[a\leq X\leq b]=\int _{a}^{b}f_{X}(x)\,dx.}</annotation>

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Hence, 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 the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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

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<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)={\frac {d}{dx}}F_{X}(x).}</annotation>

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Intuitively, one can think of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)\,dx}">

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

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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"> falling within the infinitesimal <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,x+dx]}">

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

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</math><img src="./f07271dbe3f8967834a2eaf143decd7e41c61d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.373ex; height:2.843ex;" alt="{\displaystyle [x,x+dx]}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

(This definition may be extended to any probability distribution using the <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure-theoretic</a> <a href="Probability_axioms" title="Probability axioms">definition of probability</a>.)

A <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"> with values in a <a href="Measurable_space" title="Measurable space">measurable space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}">

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

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</math><img src="./478d656144cecc9f1e5bbd8c4a14d4dd7092b82c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.622ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}" loading="lazy"> (usually <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> with the <a href="Borel_set" title="Borel set">Borel sets</a> as measurable subsets) has as <a href="Probability_distribution#Formal_definition" title="Probability distribution">probability distribution</a> the <a href="Pushforward_measure" title="Pushforward measure">pushforward measure</a> X∗P on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}">

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

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</math><img src="./478d656144cecc9f1e5bbd8c4a14d4dd7092b82c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.622ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}" loading="lazy">: the density 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"> with respect to a reference measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</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"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}">

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

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</math><img src="./478d656144cecc9f1e5bbd8c4a14d4dd7092b82c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.622ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {X}},{\mathcal {A}})}" loading="lazy"> is the <a href="Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a>:

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<annotation encoding="application/x-tex">{\displaystyle f={\frac {dX_{*}P}{d\mu }}.}</annotation>

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That is, f is any measurable function with the property that:

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<annotation encoding="application/x-tex">{\displaystyle \Pr[X\in A]=\int _{X^{-1}A}\,dP=\int _{A}f\,d\mu }</annotation>

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</math>or any measurable set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {A}}.}">

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

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</math><img src="./653501c182e4c1b8ea723b0cd7b39256e7009824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.134ex; height:2.343ex;" alt="{\displaystyle A\in {\mathcal {A}}.}" loading="lazy">

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

In the <a href="#Absolutely_continuous_univariate_distributions">continuous univariate case above</a>, the reference measure is the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>. The <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> of a <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a> is the density with respect to the <a href="Counting_measure" title="Counting measure">counting measure</a> over the sample space (usually the set of <a href="Integer" title="Integer">integers</a>, or some subset thereof).

It is not possible to define a density with reference to an arbitrary measure (e.g. one can not choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>.

<div class="mw-heading mw-heading2"><h2 id="Further_details">Further details</h2></div>

Unlike a probability, a probability density function can take on values greater than one; for example, the <a href="Continuous_uniform_distribution" title="Continuous uniform distribution">continuous uniform distribution</a> on the interval [0, 1/2] has probability density f(x) = 2 for 0 ≤ x ≤ 1/2 and f(x) = 0 elsewhere.

The <a href="Normal_distribution#Standard_normal_distribution" title="Normal distribution">standard normal distribution</a> has probability density<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\sqrt {2\pi }}}\,e^{-x^{2}/2}.}">

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

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If a random variable X is given and its distribution admits a probability density function f, then the <a href="Expected_value" title="Expected value">expected value</a> of X (if the expected value exists) can be calculated as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }x\,f(x)\,dx.}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }x\,f(x)\,dx.}</annotation>

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Not every probability distribution has a density function: the distributions of <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a> do not; nor does the <a href="Cantor_distribution" title="Cantor distribution">Cantor distribution</a>, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

A distribution has a density function if its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> F(x) is <a href="Absolute_continuity" title="Absolute continuity">absolutely continuous</a>.<a href="#cite_note-5">[5]</a> In this case: F is <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a> <a href="Derivative" title="Derivative">differentiable</a>, and its derivative can be used as probability density:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}F(x)=f(x).}">

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

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If a probability distribution admits a density, then the probability of every one-point set {a} is zero; the same holds for finite and countable sets.

Two probability densities f and g represent the same <a href="Probability_distribution" title="Probability distribution">probability distribution</a> precisely if they differ only on a set of <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a> <a href="Measure_zero" class="mw-redirect" title="Measure zero">measure zero</a>.

In the field of <a href="Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:

If dt is an infinitely small number, the probability that X is included within the interval (t, t + dt) is equal to f(t) dt, or:

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<annotation encoding="application/x-tex">{\displaystyle \Pr(t<X<t+dt)=f(t)\,dt.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Link_between_discrete_and_continuous_distributions">Link between discrete and continuous distributions</h2></div>

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a <a href="Generalized_function" title="Generalized function">generalized</a> probability density function using the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. (This is not possible with a probability density function in the sense defined above, it may be done with a <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a>.) For example, consider a binary discrete <a href="Random_variable" title="Random variable">random variable</a> having the <a href="Rademacher_distribution" title="Rademacher distribution">Rademacher distribution</a>—that is, taking −1 or 1 for values, with probability <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>1⁄2 each. The density of probability associated with this variable is:

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<annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {1}{2}}(\delta (t+1)+\delta (t-1)).}</annotation>

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More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\sum _{i=1}^{n}p_{i}\,\delta (t-x_{i}),}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}">

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

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</math><img src="./737e02a5fbf8bc31d443c91025339f9fd1de1065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}" loading="lazy"> are the discrete values accessible to the variable and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\ldots ,p_{n}}">

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

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</math><img src="./4d4021f35aca5bc2f464be2b99b0fae72f2c7870.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.879ex; height:2.009ex;" alt="{\displaystyle p_{1},\ldots ,p_{n}}" loading="lazy"> are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. The above expression allows for determining statistical characteristics of such a discrete variable (such as the <a href="Mean" title="Mean">mean</a>, <a href="Variance" title="Variance">variance</a>, and <a href="Kurtosis" title="Kurtosis">kurtosis</a>), starting from the formulas given for a continuous distribution of the probability.

<div class="mw-heading mw-heading2"><h2 id="Families_of_densities">Families of densities</h2></div>

It is common for probability density functions (and <a href="Probability_mass_function" title="Probability mass function">probability mass functions</a>) to be parametrized—that is, to be characterized by unspecified <a href="Parameter" title="Parameter">parameters</a>. For example, the <a href="Normal_distribution" title="Normal distribution">normal distribution</a> is parametrized in terms of the <a href="Mean" title="Mean">mean</a> and the <a href="Variance" title="Variance">variance</a>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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

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</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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">

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

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</math><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy"> respectively, giving the family of densities

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<annotation encoding="application/x-tex">{\displaystyle f(x;\mu ,\sigma ^{2})={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}.}</annotation>

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Different values of the parameters describe different distributions of different <a href="Random_variable" title="Random variable">random variables</a> on the same <a href="Sample_space" title="Sample space">sample space</a> (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes. A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the <a href="Normalization_factor" class="mw-redirect" title="Normalization factor">normalization factor</a> of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the <a href="Kernel_(statistics)" title="Kernel (statistics)">kernel</a> of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones.

<div class="mw-heading mw-heading2"><h2 id="Densities_associated_with_multiple_variables">Densities associated with multiple variables</h2></div>

For continuous <a href="Random_variable" title="Random variable">random variables</a> X1, ..., Xn, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the n variables, such that, for any domain D in the n-dimensional space of the values of the variables X1, ..., Xn, the probability that a realisation of the set variables falls inside the domain D is

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<annotation encoding="application/x-tex">{\displaystyle \Pr \left(X_{1},\ldots ,X_{n}\in D\right)=\int _{D}f_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})\,dx_{1}\cdots dx_{n}.}</annotation>

</semantics>

</math>

If F(x1, ..., xn) = Pr(X1 ≤ x1, ..., Xn ≤ xn) is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the vector (X1, ..., Xn), then the joint probability density function can be computed as a partial derivative

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\left.{\frac {\partial ^{n}F}{\partial x_{1}\cdots \partial x_{n}}}\right|_{x}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Marginal_densities">Marginal densities</h3></div>

For i = 1, 2, ..., n, let fXi(xi) be the probability density function associated with variable Xi alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables X1, ..., Xn by integrating over all values of the other n − 1 variables:

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<annotation encoding="application/x-tex">{\displaystyle f_{X_{i}}(x_{i})=\int f(x_{1},\ldots ,x_{n})\,dx_{1}\cdots dx_{i-1}\,dx_{i+1}\cdots dx_{n}.}</annotation>

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

Continuous random variables X1, ..., Xn admitting a joint density are all <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> from each other if

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<annotation encoding="application/x-tex">{\displaystyle f_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=f_{X_{1}}(x_{1})\cdots f_{X_{n}}(x_{n}).}</annotation>

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

If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable

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

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(where each fi is not necessarily a density) then the n variables in the set are all <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> from each other, and the marginal probability density function of each of them is given by

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<annotation encoding="application/x-tex">{\displaystyle f_{X_{i}}(x_{i})={\frac {f_{i}(x_{i})}{\int f_{i}(x)\,dx}}.}</annotation>

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

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {R}}}">

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

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</math><img src="./f361eec9da18669c0e7e869c57bae6c657f53522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {R}}}" loading="lazy"> a 2-dimensional random vector of coordinates (X, Y): the probability to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {R}}}">

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

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<annotation encoding="application/x-tex">{\displaystyle \Pr \left(X>0,Y>0\right)=\int _{0}^{\infty }\int _{0}^{\infty }f_{X,Y}(x,y)\,dx\,dy.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Function_of_random_variables_and_change_of_variables_in_the_probability_density_function">Function of random variables and change of variables in the probability density function</h2></div>

If the probability density function of a random variable (or vector) X is given as fX(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable Y = g(X). This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape fg(X) = fY using a known (for instance, uniform) random number generator.

It is tempting to think that in order to find the expected value E(g(X)), one must first find the probability density fg(X) of the new random variable Y = g(X). However, rather than computing

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} {\big (}g(X){\big )}=\int _{-\infty }^{\infty }yf_{g(X)}(y)\,dy,}</annotation>

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one may find instead

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} {\big (}g(X){\big )}=\int _{-\infty }^{\infty }g(x)f_{X}(x)\,dx.}</annotation>

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The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a <a href="One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one function</a>. In some cases the latter integral is computed much more easily than the former. See <a href="Law_of_the_unconscious_statistician" title="Law of the unconscious statistician">Law of the unconscious statistician</a>.

<div class="mw-heading mw-heading3"><h3 id="Scalar_to_scalar">Scalar to scalar</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} \to \mathbb {R} }">

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

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</math><img src="./bdfd1e16b7f932cdc2716a1b6bbe345089b250cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.023ex; height:2.509ex;" alt="{\displaystyle g:\mathbb {R} \to \mathbb {R} }" loading="lazy"> be a <a href="Monotonic_function" title="Monotonic function">monotonic function</a>, then the resulting density function is<a href="#cite_note-6">[6]</a>

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

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Here g−1 denotes the <a href="Inverse_function" title="Inverse function">inverse function</a>.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,

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<annotation encoding="application/x-tex">{\displaystyle \left|f_{Y}(y)\,dy\right|=\left|f_{X}(x)\,dx\right|,}</annotation>

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

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For functions that are not monotonic, the probability density function for y is

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n(y)}\left|{\frac {d}{dy}}g_{k}^{-1}(y)\right|\cdot f_{X}{\big (}g_{k}^{-1}(y){\big )},}</annotation>

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where n(y) is the number of solutions in x for the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=y}">

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<div class="mw-heading mw-heading3"><h3 id="Vector_to_vector">Vector to vector</h3></div>

Suppose x is an n-dimensional random variable with joint density f. If y = G(x), where G is a <a href="Bijective" class="mw-redirect" title="Bijective">bijective</a>, <a href="Differentiable_function" title="Differentiable function">differentiable function</a>, then y has density pY:

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<annotation encoding="application/x-tex">{\displaystyle p_{Y}(\mathbf {y} )=f{\Bigl (}G^{-1}(\mathbf {y} ){\Bigr )}\left|\det \left[\left.{\frac {dG^{-1}(\mathbf {z} )}{d\mathbf {z} }}\right|_{\mathbf {z} =\mathbf {y} }\right]\right|}</annotation>

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with the differential regarded as the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of the inverse of G(⋅), evaluated at y.<a href="#cite_note-7">[7]</a>

For example, in the 2-dimensional case x = (x1, x2), suppose the transform G is given as y1 = G1(x1, x2), y2 = G2(x1, x2) with inverses x1 = G1−1(y1, y2), x2 = G2−1(y1, y2). The joint distribution for y = (y1, y2) has density<a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle p_{Y_{1},Y_{2}}(y_{1},y_{2})=f_{X_{1},X_{2}}{\big (}G_{1}^{-1}(y_{1},y_{2}),G_{2}^{-1}(y_{1},y_{2}){\big )}\left\vert {\frac {\partial G_{1}^{-1}}{\partial y_{1}}}{\frac {\partial G_{2}^{-1}}{\partial y_{2}}}-{\frac {\partial G_{1}^{-1}}{\partial y_{2}}}{\frac {\partial G_{2}^{-1}}{\partial y_{1}}}\right\vert .}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Vector_to_scalar">Vector to scalar</h3></div>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./0e39d5850ef2fee687b6488152873b0d6f7733fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.913ex; height:2.343ex;" alt="{\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"> be a differentiable function and <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"> be a random vector taking values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

<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"> be the probability density function 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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\cdot )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7b74d4bfd8b80a42cd8184a350f64fad0faa7adc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.505ex; height:2.843ex;" alt="{\displaystyle \delta (\cdot )}" loading="lazy"> be the <a href="Dirac_delta" class="mw-redirect" title="Dirac delta">Dirac delta</a> function. It is possible to use the formulas above to determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6e8b3fed56c8af1f38961f6e4ec0d64fe50ecb4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.625ex; height:2.509ex;" alt="{\displaystyle f_{Y}}" loading="lazy">, the probability density function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=V(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1495a2bccd7b034e591c2929667b89db573bb6bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.448ex; height:2.843ex;" alt="{\displaystyle Y=V(X)}" loading="lazy">, which will be given by

<msub>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{Y}(y)=\int _{\mathbb {R} ^{n}}f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\big )}\,d\mathbf {x} .}</annotation>

</semantics>

</math>

This result leads to the <a href="Law_of_the_unconscious_statistician" title="Law of the unconscious statistician">law of the unconscious statistician</a>:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

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</msub>

<msub>

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<mtr>

<mtd>

<msub>

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</msub>

<msub>

<msup>

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</mrow>

</msup>

</mrow>

</msub>

<msub>

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</msub>

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</mtr>

<mtr>

<mtd>

<msub>

<msup>

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</mrow>

</msup>

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</msub>

<msub>

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</msub>

<msub>

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</mtr>

<mtr>

<mtd>

<msub>

<msup>

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</msup>

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</msub>

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<msub>

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</msub>

</mrow>

</mrow>

<msub>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} _{Y}[Y]&=\int _{\mathbb {R} }yf_{Y}(y)\,dy\\&=\int _{\mathbb {R} }y\int _{\mathbb {R} ^{n}}f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\big )}\,d\mathbf {x} \,dy\\&=\int _{{\mathbb {R} }^{n}}\int _{\mathbb {R} }yf_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\big )}\,dy\,d\mathbf {x} \\&=\int _{\mathbb {R} ^{n}}V(\mathbf {x} )f_{X}(\mathbf {x} )\,d\mathbf {x} =\operatorname {E} _{X}[V(X)].\end{aligned}}}</annotation>

</semantics>

</math>

Proof:

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> be a collapsed random variable with probability density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{Z}(z)=\delta (z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{Z}(z)=\delta (z)}</annotation>

</semantics>

</math><img src="./af80c337bdbe9f42e5f48af63d52216975be1531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.621ex; height:2.843ex;" alt="{\displaystyle p_{Z}(z)=\delta (z)}" loading="lazy"> (i.e., a constant equal to zero). Let the random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {X}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./19b6c2d2aa76b9cf010d897dc2ce988acf539624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.676ex;" alt="{\displaystyle {\tilde {X}}}" loading="lazy"> and the transform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(Z,X)={\begin{bmatrix}Z+V(X)\\X\end{bmatrix}}={\begin{bmatrix}Y\\{\tilde {X}}\end{bmatrix}}.}</annotation>

</semantics>

</math>

It is clear that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> is a bijective mapping, and the Jacobian of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8a866bed9de4a9cf9bca4eaf753811839f2b5ad5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.436ex; height:2.676ex;" alt="{\displaystyle H^{-1}}" loading="lazy"> is given by:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<mrow>

<mover>

</mrow>

</mover>

</mrow>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<mrow>

<mover>

</mrow>

</mover>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

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</msub>

</mtd>

</mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dH^{-1}(y,{\tilde {\mathbf {x} }})}{dy\,d{\tilde {\mathbf {x} }}}}={\begin{bmatrix}1&-{\frac {dV({\tilde {\mathbf {x} }})}{d{\tilde {\mathbf {x} }}}}\\\mathbf {0} _{n\times 1}&\mathbf {I} _{n\times n}\end{bmatrix}},}</annotation>

</semantics>

</math>

which is an upper <a href="Triangular_matrix" title="Triangular matrix">triangular matrix</a> with ones on the <a href="Main_diagonal" title="Main diagonal">main diagonal</a>, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f_{Y,X}(y,x)=f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\big )},}</annotation>

</semantics>

</math>

which if marginalized over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Sums_of_independent_random_variables">Sums of independent random variables</h2></div>

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

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="List_of_convolutions_of_probability_distributions" title="List of convolutions of probability distributions">List of convolutions of probability distributions</a></div>

The probability density function of the sum of two <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> random variables U and V, each of which has a probability density function, is the <a href="Convolution" title="Convolution">convolution</a> of their separate density functions:

<msub>

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</msub>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{U+V}(x)=\int _{-\infty }^{\infty }f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}\right)(x)}</annotation>

</semantics>

</math>

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U1, ..., UN:

<msub>

<msub>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle f_{U_{1}+\cdots +U}(x)=\left(f_{U_{1}}*\cdots *f_{U_{N}}\right)(x)}</annotation>

</semantics>

</math>

This can be derived from a two-way change of variables involving Y = U + V and Z = V, similarly to the example below for the quotient of independent random variables.

<div class="mw-heading mw-heading2"><h2 id="Products_and_quotients_of_independent_random_variables">Products and quotients of independent random variables</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Product_distribution" class="mw-redirect" title="Product distribution">Product distribution</a> and <a href="Ratio_distribution" title="Ratio distribution">Ratio distribution</a></div>

Given two independent random variables U and V, each of which has a probability density function, the density of the product Y = UV and quotient Y = U/V can be computed by a change of variables.

<div class="mw-heading mw-heading3"><h3 id="Example:_Quotient_distribution">Example: Quotient distribution</h3></div>

To compute the quotient Y = U/V of two independent random variables U and V, define the following transformation:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y&=U/V\\[1ex]Z&=V\end{aligned}}}</annotation>

</semantics>

</math>

Then, the joint density p(y,z) can be computed by a change of variables from U,V to Y,Z, and Y can be derived by <a href="Marginalizing_out" class="mw-redirect" title="Marginalizing out">marginalizing out</a> Z from the joint density.

The inverse transformation is

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U&=YZ\\V&=Z\end{aligned}}}</annotation>

</semantics>

</math>

The absolute value of the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> determinant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(U,V\mid Y,Z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(U,V\mid Y,Z)}</annotation>

</semantics>

</math><img src="./62010573d050ae3a3fae1e8e9bf339716478e7f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.309ex; height:2.843ex;" alt="{\displaystyle J(U,V\mid Y,Z)}" loading="lazy"> of this transformation is:

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<mtr>

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<annotation encoding="application/x-tex">{\displaystyle \left|\det {\begin{bmatrix}{\frac {\partial u}{\partial y}}&{\frac {\partial u}{\partial z}}\\{\frac {\partial v}{\partial y}}&{\frac {\partial v}{\partial z}}\end{bmatrix}}\right|=\left|\det {\begin{bmatrix}z&y\\0&1\end{bmatrix}}\right|=|z|.}</annotation>

</semantics>

</math>

Thus:

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle p(y,z)=p(u,v)\,J(u,v\mid y,z)=p(u)\,p(v)\,J(u,v\mid y,z)=p_{U}(yz)\,p_{V}(z)\,|z|.}</annotation>

</semantics>

</math>

And the distribution of Y can be computed by <a href="Marginalizing_out" class="mw-redirect" title="Marginalizing out">marginalizing out</a> Z:

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle p(y)=\int _{-\infty }^{\infty }p_{U}(yz)\,p_{V}(z)\,|z|\,dz}</annotation>

</semantics>

</math>

This method crucially requires that the transformation from U,V to Y,Z be <a href="Bijective" class="mw-redirect" title="Bijective">bijective</a>. The above transformation meets this because Z can be mapped directly back to V, and for a given V the quotient U/V is <a href="Monotonic" class="mw-redirect" title="Monotonic">monotonic</a>. This is similarly the case for the sum U + V, difference U − V and product UV.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.

<div class="mw-heading mw-heading3"><h3 id="Example:_Quotient_of_two_standard_normals">Example: Quotient of two standard normals</h3></div>

Given two <a href="Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal</a> variables U and V, the quotient can be computed as follows. First, the variables have the following density functions:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(u)&={\frac {1}{\sqrt {2\pi }}}e^{-{u^{2}}/{2}}\\[1ex]p(v)&={\frac {1}{\sqrt {2\pi }}}e^{-{v^{2}}/{2}}\end{aligned}}}</annotation>

</semantics>

</math>

We transform as described above:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Y&=U/V\\[1ex]Z&=V\end{aligned}}}</annotation>

</semantics>

</math>

This leads to:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(y)&=\int _{-\infty }^{\infty }p_{U}(yz)\,p_{V}(z)\,|z|\,dz\\[5pt]&=\int _{-\infty }^{\infty }{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}y^{2}z^{2}}{\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}z^{2}}|z|\,dz\\[5pt]&=\int _{-\infty }^{\infty }{\frac {1}{2\pi }}e^{-{\frac {1}{2}}\left(y^{2}+1\right)z^{2}}|z|\,dz\\[5pt]&=2\int _{0}^{\infty }{\frac {1}{2\pi }}e^{-{\frac {1}{2}}\left(y^{2}+1\right)z^{2}}z\,dz\\[5pt]&=\int _{0}^{\infty }{\frac {1}{\pi }}e^{-\left(y^{2}+1\right)u}\,du&&u={\tfrac {1}{2}}z^{2}\\[5pt]&=\left.-{\frac {1}{\pi \left(y^{2}+1\right)}}e^{-\left(y^{2}+1\right)u}\right|_{u=0}^{\infty }\\[5pt]&={\frac {1}{\pi \left(y^{2}+1\right)}}\end{aligned}}}</annotation>

</semantics>

</math>

This is the density of a standard <a href="Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>.

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a> – Estimate of an unobservable underlying probability density function</li>

<li><a href="Kernel_density_estimation" title="Kernel density estimation">Kernel density estimation</a> – EstimatorPages displaying short descriptions with no spaces</li>

<li><a href="Likelihood_function" title="Likelihood function">Likelihood function</a> – Function related to statistics and probability theory</li>

<li><a href="List_of_probability_distributions" title="List of probability distributions">List of probability distributions</a></li>

<li><a href="Probability_amplitude" title="Probability amplitude">Probability amplitude</a> – Complex number whose squared absolute value is a probability</li>

<li><a href="Probability_mass_function" title="Probability mass function">Probability mass function</a> – Discrete-variable probability distribution</li>

<li><a href="Secondary_measure" title="Secondary measure">Secondary measure</a> – Concept in mathematics</li>

<li><a href="Coalescence_(statistics)" title="Coalescence (statistics)"> Merging independent probability density functions</a></li>

<li>Uses as position probability density:

<ul><li><a href="Atomic_orbital" title="Atomic orbital">Atomic orbital</a> – Function describing an electron in an atom</li>

<li><a href="Home_range" title="Home range">Home range</a> – Area in which an animal lives and moves</li></ul></li></ul>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions">"AP Statistics Review - Density Curves and the Normal Distributions"</a>. Archived from <a rel="nofollow" class="external text" href="https://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions">the original</a> on 2 April 2015. Retrieved 16 March 2015.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFGrinsteadSnell2009" class="citation book cs1">Grinstead, Charles M.; Snell, J. Laurie (2009). <a rel="nofollow" class="external text" href="https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf">"Conditional Probability - Discrete Conditional"</a> (PDF). Grinstead & Snell's Introduction to Probability. Orange Grove Texts. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1616100469</bdi>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20030425090244/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf">Archived</a> (PDF) from the original on 2003-04-25. Retrieved 2019-07-25.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stats.stackexchange.com/q/541479">"probability - Is a uniformly random number over the real line a valid distribution?"</a>. Cross Validated. Retrieved 2021-10-06.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Ord, J.K. (1972) Families of Frequency Distributions, Griffin. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-85264-137-0</bdi> (for example, Table 5.1 and Example 5.4)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFScalas2025" class="citation book cs1">Scalas, Enrico (2025). <a rel="nofollow" class="external text" href="https://www.lem.sssup.it/phd/documents/probpisanew.pdf">Introduction to Probability Theory for Economists</a> (PDF). self-published. p. 28. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241210231824/https://www.lem.sssup.it/phd/documents/probpisanew.pdf">Archived</a> (PDF) from the original on Dec 10, 2024. Retrieved July 30, 2025.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFSiegrist2020" class="citation web cs1">Siegrist, Kyle (5 May 2020). <a rel="nofollow" class="external text" href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables#The_Change_of_Variables_Formula">"Transformations of Random Variables"</a>. LibreTexts Statistics. Retrieved 22 December 2023.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFDevoreBerk2007" class="citation book cs1">Devore, Jay L.; Berk, Kenneth N. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3X7Qca6CcfkC&pg=PA263">Modern Mathematical Statistics with Applications</a>. Cengage. p. 263. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-40473-4</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFDavid2007" class="citation book cs1">David, Stirzaker (2007-01-01). Elementary Probability. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0521534284</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/851313783">851313783</a>.</cite>

</li>

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

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBillingsley1979" class="citation book cs1"><a href="Patrick_Billingsley" title="Patrick Billingsley">Billingsley, Patrick</a> (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-00710-2</bdi>.</cite></li>

<li><cite id="CITEREFCasellaBerger2002" class="citation book cs1"><a href="George_Casella" title="George Casella">Casella, George</a>; <a href="Roger_Lee_Berger" title="Roger Lee Berger">Berger, Roger L.</a> (2002). Statistical Inference (Second ed.). Thomson Learning. pp. 34–37. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-24312-6</bdi>.</cite></li>

<li><cite id="CITEREFStirzaker2003" class="citation book cs1">Stirzaker, David (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/elementaryprobab0000stir">Elementary Probability</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-42028-8</bdi>.</cite> Chapters 7 to 9 are about continuous variables.</li></ul>

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

<ul><li><cite id="CITEREFUshakov2001" class="citation cs2">Ushakov, N.G. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Density_of_a_probability_distribution">"Density of a probability distribution"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ProbabilityDensityFunction.html">"Probability density function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></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="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (cdf)</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><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-30" href="https://en.wikipedia.org/wiki/?title=Probability_density_function&oldid=1303334817">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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