View source for Physics:Probability density function

{{Short description|Function whose integral over a region describes the probability of an event occurring in that region}}
[[Image:Boxplot vs PDF.svg|thumb|350px|[[Box plot|Box plot]] and probability density function of a [[Normal distribution|normal distribution]] {{math|''N''(0,&thinsp;''σ''<sup>2</sup>)}}.]]
[[Image:visualisation_mode_median_mean.svg|thumb|150px|Geometric visualisation of the [[Mode (statistics)|mode]], median and mean of an arbitrary unimodal probability density function.<ref>{{cite web|title=AP Statistics Review - Density Curves and the Normal Distributions|url=http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions | access-date=16 March 2015| archive-url=https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions | archive-date=2 April 2015| url-status=dead}}</ref>]]
{{citations needed|date=June 2022}}
In [[Probability theory|probability theory]], a '''probability density function''' ('''PDF'''), '''density function''', or '''density''' of an absolutely continuous random variable, is a [[Function (mathematics)|function]] whose value at any given sample (or point) in the [[Sample space|sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''[[Relative likelihood|relative likelihood]]'' that the value of the random variable would be equal to that sample.<ref>{{cite book| chapter-url=https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf |archive-url=https://web.archive.org/web/20030425090244/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf |archive-date=2003-04-25 |url-status=live| chapter=Conditional Probability - Discrete Conditional| last1=Grinstead|first1=Charles M.| last2=Snell|first2=J. Laurie| publisher=Orange Grove Texts| isbn=978-1616100469 | title=Grinstead & Snell's Introduction to Probability| date=2009| access-date=2019-07-25}}</ref><ref>{{Cite web|title=probability - Is a uniformly random number over the real line a valid distribution?| url=https://stats.stackexchange.com/q/541479 |access-date=2021-10-06| website=Cross Validated}}</ref> Probability density  is the  probability per unit length,  in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), 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.

In a more precise sense, the PDF is used to specify the probability of the [[Random variable|random variable]] falling ''within a particular range of values'', as opposed to taking on any one value. This probability is given by the [[Integral|integral]] of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1.

The terms ''probability distribution function'' and ''probability function'' have also sometimes been used to 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 [[Probability distribution|probability distribution]] is defined as a function over general sets of values or it may refer to the [[Cumulative distribution function|cumulative distribution function]], or it may be a [[Probability mass function|probability mass function]] (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion.<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. {{isbn|0-85264-137-0}} (for example, Table 5.1 and Example 5.4)</ref> In general though, the PMF is used in the context of [[Continuous or discrete variable#Discrete variable|discrete random variables]] (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

==Example==

[[File:4 continuous probability density functions.png|thumb|Examples of four continuous probability density functions.]]

Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives {{em|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 dying during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour<sup>−1</sup>). 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<sup>−1</sup>. This quantity 2 hour<sup>−1</sup> 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<sup>−1</sup>) ''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<sup>−1</sup>)×(1 nanosecond) ≈ {{val|6e-13}} (using the [[Conversion of units|unit conversion]] {{val|3.6e12}} nanoseconds = 1 hour).

There is a probability density function ''f'' with ''f''(5 hours) = 2 hour<sup>−1</sup>. The [[Integral|integral]] 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.

==Absolutely continuous univariate distributions==
A probability density function is most commonly associated with absolutely continuous [[Univariate distribution|univariate distribution]]s. A [[Random variable|random variable]] <math>X</math> has density <math>f_X</math>, where <math>f_X</math> is a non-negative [[Lebesgue integration|Lebesgue-integrable]] function, if:
<math display="block"> \Pr [a \le X \le b] = \int_a^b f_X(x) \, dx .</math>

Hence, if <math>F_X</math> is the [[Cumulative distribution function|cumulative distribution function]] of <math>X</math>, then:
<math display="block">F_X(x) = \int_{-\infty}^x f_X(u) \, du ,</math>
and (if <math>f_X</math> is continuous at <math>x</math>)
<math display="block"> f_X(x) = \frac{d}{dx} F_X(x) .</math>

Intuitively, one can think of <math>f_X(x) \, dx</math> as being the probability of <math>X</math> falling within the infinitesimal [[Interval (mathematics)|interval]] <math>[x,x+dx]</math>.

==Formal definition==
(''This definition may be extended to any probability distribution using the measure-theoretic [[Probability axioms|definition of probability]].'')

A [[Random variable|random variable]] <math>X</math> with values in a [[Measurable space|measurable space]] <math>(\mathcal{X}, \mathcal{A})</math> (usually <math>\mathbb{R}^n</math> with the [[Borel set]]s as measurable subsets) has as [[Probability distribution#Formal definition|probability distribution]] the measure ''X''<sub>∗</sub>''P'' on <math>(\mathcal{X}, \mathcal{A})</math>: the '''density''' of <math>X</math> with respect to a reference measure <math>\mu</math> on <math>(\mathcal{X}, \mathcal{A})</math> is the Radon–Nikodym derivative:
<math display="block">f = \frac{dX_*P}{d\mu} .</math>

That is, ''f'' is any measurable function with the property that:
<math display="block">\Pr [X \in A ] = \int_{X^{-1} A} \, dP = \int_A f \, d\mu </math>
for any measurable set <math>A \in \mathcal{A}.</math>

===Discussion===
In the continuous univariate case above, the reference measure is the [[Lebesgue measure]]. The [[Probability mass function|probability mass function]] of a discrete random variable is the density with respect to the [[Counting measure|counting measure]] over the sample space (usually the set of [[Integer|integer]]s, 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 [[Almost everywhere|almost everywhere]].

==Further details==
Unlike a probability, a probability density function can take on values greater than one; for example, the [[Continuous uniform distribution|continuous uniform distribution]] on the interval {{closed-closed|0, 1/2}} has probability density {{math|1=''f''(''x'') = 2}} for {{math|0 ≤ ''x'' ≤ 1/2}} and {{math|1=''f''(''x'') = 0}} elsewhere.

The [[Normal distribution#Standard normal distribution|standard normal distribution]] has probability density
<math display="block">f(x) = \frac{1}{\sqrt{2\pi}}\, e^{-x^2/2}.</math>

If a random variable {{math|''X''}} is given and its distribution admits a probability density function {{math|''f''}}, then the [[Expected value|expected value]] of {{math|''X''}} (if the expected value exists) can be calculated as
<math display="block">\operatorname{E}[X] = \int_{-\infty}^\infty x\,f(x)\,dx.</math>

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the [[Cantor distribution]], 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 and only if its [[Cumulative distribution function|cumulative distribution function]] {{math|''F''(''x'')}} is [[Absolute continuity|absolutely continuous]]. In this case: {{math|''F''}} is [[Almost everywhere|almost everywhere]] [[Derivative|differentiable]], and its derivative can be used as probability density:
<math display="block">\frac{d}{dx}F(x) = f(x).</math>

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

Two probability densities {{math|''f''}} and {{math|''g''}} represent the same [[Probability distribution|probability distribution]] precisely if they differ only on a set of [[Lebesgue measure|Lebesgue]] measure zero.

In the field of [[Physics:Statistical physics|statistical physics]], a non-formal reformulation of the relation above between the derivative of the [[Cumulative distribution function|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 {{math|''dt''}} is an infinitely small number, the probability that {{math|''X''}} is included within the interval {{open-open|''t'', ''t'' + ''dt''}} is equal to {{math|''f''(''t'') ''dt''}}, or:
<math display="block">\Pr(t<X<t+dt) = f(t)\,dt.</math>

==Link between discrete and continuous distributions==

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a [[Generalized function|generalized]] probability density function using the [[Dirac delta function]]. (This is not possible with a probability density function in the sense defined above, it may be done with a [[Distribution (mathematics)|distribution]].) For example, consider a binary discrete [[Random variable|random variable]] having the [[Rademacher distribution]]—that is, taking −1 or 1 for values, with probability {{1/2}} each. The density of probability associated with this variable is:
<math display="block">f(t) = \frac{1}{2} (\delta(t+1)+\delta(t-1)).</math>

More generally, if a discrete variable can take {{mvar|n}} different values among real numbers, then the associated probability density function is:
<math display="block">f(t) = \sum_{i=1}^n p_i\, \delta(t-x_i),</math>
where <math>x_1, \ldots, x_n</math> are the discrete values accessible to the variable and <math>p_1, \ldots, p_n</math> 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 [[Mean|mean]], [[Variance|variance]], and [[Kurtosis|kurtosis]]), starting from the formulas given for a continuous distribution of the probability.

== Families of densities ==
It is common for probability density functions (and [[Probability mass function|probability mass function]]s) to be parametrized—that is, to be characterized by unspecified [[Parameter|parameter]]s. For example, the [[Normal distribution|normal distribution]] is parametrized in terms of the [[Mean|mean]] and the [[Variance|variance]], denoted by <math>\mu</math> and <math>\sigma^2</math> respectively, giving the family of densities
<math display="block">
   f(x;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }.
</math>
Different values of the parameters describe different distributions of different [[Random variable|random variable]]s on the same [[Sample space|sample space]] (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 normalization factor 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 [[Kernel (statistics)|kernel]] 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.

== Densities associated with multiple variables ==<!-- This section is linked from Sufficiency (statistics) -->

For continuous [[Random variable|random variable]]s {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, 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 {{mvar|n}} variables, such that, for any domain {{mvar|D}} in the {{mvar|n}}-dimensional space of the values of the variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}}, the probability that a realisation of the set variables falls inside the domain {{mvar|D}} is
<math display="block">\Pr \left( X_1,\ldots,X_n \isin D \right)
 = \int_D f_{X_1,\ldots,X_n}(x_1,\ldots,x_n)\,dx_1 \cdots dx_n.</math>

If {{math|1=''F''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = Pr(''X''<sub>1</sub> ≤ ''x''<sub>1</sub>, ..., ''X''<sub>''n''</sub> ≤ ''x''<sub>''n''</sub>)}} is the [[Cumulative distribution function|cumulative distribution function]] of the vector {{math|(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}}, then the joint probability density function can be computed as a partial derivative
<math display="block"> f(x) = \left.\frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \right|_x </math>

===Marginal densities===
For {{math|1=''i'' = 1, 2, ..., ''n''}}, let {{math|''f''<sub>''X''<sub>''i''</sub></sub>(''x''<sub>''i''</sub>)}} be the probability density function associated with variable {{math|''X<sub>i</sub>''}} alone.  This is called the marginal density function, and can be deduced from the probability density associated with the random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} by integrating over all values of the other {{math|''n'' − 1}} variables:
<math display="block">f_{X_i}(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_{i-1}\,dx_{i+1}\cdots dx_n .</math>

===Independence===

Continuous random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} admitting a joint density are all independent from each other if and only if
<math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_{X_1}(x_1)\cdots f_{X_n}(x_n).</math>

===Corollary===

If the joint probability density function of a vector of {{mvar|n}} random variables can be factored into a product of {{mvar|n}} functions of one variable
<math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n),</math>
(where each {{math|''f<sub>i</sub>''}} is not necessarily a density) then the {{mvar|n}} variables in the set are all independent from each other, and the marginal probability density function of each of them is given by
<math display="block">f_{X_i}(x_i) = \frac{f_i(x_i)}{\int f_i(x)\,dx}.</math>

===Example===

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>\vec R</math> a 2-dimensional random vector of coordinates {{math|(''X'', ''Y'')}}: the probability to obtain <math>\vec R</math> in the quarter plane of positive {{math|''x''}} and {{math|''y''}} is
<math display="block">\Pr \left( X > 0, Y > 0 \right)
 = \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.</math>

==Function of random variables and change of variables in the probability density function==

If the probability density function of a random variable (or vector) {{math|''X''}} is given as {{math|''f<sub>X</sub>''(''x'')}}, it is possible (but often not necessary; see below) to calculate the probability density function of some variable {{math|1=''Y'' = ''g''(''X'')}}. This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape {{math|1=''f''<sub>''g''(''X'')</sub> = ''f<sub>Y</sub>''}} using a known (for instance, uniform) random number generator.

It is tempting to think that in order to find the expected value {{math|E(''g''(''X''))}}, one must first find the probability density {{math|''f''<sub>''g''(''X'')</sub>}} of the new random variable {{math|1=''Y'' = ''g''(''X'')}}.  However, rather than computing
<math display="block">\operatorname E\big(g(X)\big) = \int_{-\infty}^\infty y f_{g(X)}(y)\,dy, </math>
one may find instead
<math display="block">\operatorname E\big(g(X)\big) = \int_{-\infty}^\infty g(x) f_X(x)\,dx.</math>

The values of the two integrals are the same in all cases in which both {{math|''X''}} and {{math|''g''(''X'')}} actually have probability density functions.  It is not necessary that {{math|''g''}} be a one-to-one function.  In some cases the latter integral is computed much more easily than the former. See [[Law of the unconscious statistician]].

===Scalar to scalar===

Let <math> g: \Reals \to \Reals</math> be a [[Monotonic function|monotonic function]], then the resulting density function is<ref>{{cite web |last1=Siegrist |first1=Kyle |title=Transformations of Random Variables |url=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 |publisher=LibreTexts Statistics |access-date=22 December 2023}}</ref>
<math display="block">f_Y(y) = f_X\big(g^{-1}(y)\big) \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right|.</math>

Here {{math|''g''<sup>−1</sup>}} denotes the [[Inverse function|inverse function]].

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,
<math display="block">\left| f_Y(y)\, dy \right| = \left| f_X(x)\, dx \right|,</math>
or
<math display="block">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) .</math>

For functions that are not monotonic, the probability density function for {{mvar|y}} is
<math display="block">\sum_{k=1}^{n(y)} \left| \frac{d}{dy} g^{-1}_{k}(y) \right| \cdot f_X\big(g^{-1}_{k}(y)\big),</math>
where {{math|''n''(''y'')}} is the number of solutions in {{mvar|x}} for the equation <math>g(x) = y</math>, and <math>g_k^{-1}(y)</math> are these solutions.

===Vector to vector===

Suppose {{math|'''x'''}} is an {{mvar|n}}-dimensional random variable with joint density {{math|''f''}}. If {{math|1='''''y''''' = ''G''('''''x''''')}}, where {{math|''G''}} is a bijective, [[Differentiable function|differentiable function]], then {{math|'''''y'''''}} has density {{math|{{ math | ''p''<sub>'''''Y'''''</sub>}}}}:
<math display="block"> 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|</math>
with the differential regarded as the [[Jacobian matrix and determinant|Jacobian]] of the inverse of {{math|''G''(⋅)}}, evaluated at {{math|'''''y'''''}}.<ref>{{cite book |first1=Jay L. |last1=Devore |first2=Kenneth N. |last2=Berk |title=Modern Mathematical Statistics with Applications |publisher=Cengage |year=2007 |isbn=978-0-534-40473-4 |page=263 |url=https://books.google.com/books?id=3X7Qca6CcfkC&pg=PA263 }}</ref>

For example, in the 2-dimensional case {{math|1='''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>)}}, suppose the transform {{math|''G''}} is given as {{math|1=''y''<sub>1</sub> = ''G''<sub>1</sub>(''x''<sub>1</sub>, ''x''<sub>2</sub>)}}, {{math|1=''y''<sub>2</sub> = ''G''<sub>2</sub>(''x''<sub>1</sub>, ''x''<sub>2</sub>)}} with inverses {{math|1=''x''<sub>1</sub> = ''G''<sub>1</sub><sup>−1</sup>(''y''<sub>1</sub>, ''y''<sub>2</sub>)}}, {{math|1=''x''<sub>2</sub> = ''G''<sub>2</sub><sup>−1</sup>(''y''<sub>1</sub>, ''y''<sub>2</sub>)}}.  The joint distribution for '''y'''&nbsp;= (''y''<sub>1</sub>,&nbsp;y<sub>2</sub>) has density<ref>{{Cite book |title=Elementary Probability |last=David |first=Stirzaker |date=2007-01-01 |publisher=Cambridge University Press |isbn=978-0521534284 |oclc=851313783}}</ref>
<math display="block">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.</math>

===Vector to scalar===

Let <math> V: \R^n \to \R </math> be a differentiable function and <math> X </math> be a random vector taking values in <math> \R^n </math>, <math> f_X </math> be the probability density function of <math> X </math> and <math> \delta(\cdot) </math>  be the Dirac delta function. It is possible to use the formulas above to determine <math> f_Y </math>, the probability density function of <math> Y = V(X) </math>, which will be given by
<math display="block">f_Y(y) = \int_{\R^n} f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x}.</math>

This result leads to the [[Law of the unconscious statistician|law of the unconscious statistician]]:
<math display="block">\operatorname{E}_Y[Y] =\int_{\R} y f_Y(y) \, dy = \int_{\R} y \int_{\R^n} f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x} \,dy = \int_{{\mathbb R}^n} \int_{\mathbb R} y f_{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)].</math>

''Proof:''

Let <math>Z</math> be a collapsed random variable with probability density function <math>p_Z(z) = \delta(z)</math> (i.e., a constant equal to zero). Let the random vector <math>\tilde{X}</math> and the transform <math>H</math> be defined as
<math display="block">H(Z,X)=\begin{bmatrix} Z+V(X)\\ X\end{bmatrix}=\begin{bmatrix} Y\\ \tilde{X}\end{bmatrix}.</math>

It is clear that <math>H</math> is a bijective mapping, and the Jacobian of <math>H^{-1}</math> is given by:
<math display="block">\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\times1} & \mathbf{I}_{n\times n} \end{bmatrix},</math>
which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that
<math display="block">f_{Y,X}(y,x) = f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big),</math>
which if marginalized over <math>x</math> leads to the desired probability density function.

==Sums of independent random variables==

The probability density function of the sum of two independent random variables {{math|''U''}} and {{math|''V''}}, each of which has a probability density function, is the [[Convolution|convolution]] of their separate density functions:
<math display="block">
f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy
= \left( f_{U} * f_{V} \right) (x)
</math>

It is possible to generalize the previous relation to a sum of N independent random variables, with densities {{math|''U''<sub>1</sub>, ..., ''U<sub>N</sub>''}}:
<math display="block">f_{U_1 + \cdots + U_N}(x)
= \left( f_{U_1} * \cdots * f_{U_N} \right) (x)</math>

This can be derived from a two-way change of variables involving {{math|1=''Y'' = ''U'' + ''V''}} and {{math|1=''Z'' = ''V''}}, similarly to the example below for the quotient of independent random variables.

==Products and quotients of independent random variables==

Given two independent random variables {{math|''U''}} and {{math|''V''}}, each of which has a probability density function, the density of the product {{math|1=''Y'' = ''UV''}} and quotient {{math|1=''Y'' = ''U''/''V''}} can be computed by a change of variables.

===Example: Quotient distribution===
To compute the quotient {{math|1=''Y'' = ''U''/''V''}} of two independent random variables {{math|''U''}} and {{math|''V''}}, define the following transformation:
<math display="block">\begin{align}
Y &= U/V \\[1ex]
Z &= V
\end{align}</math>

Then, the joint density {{math|''p''(''y'',''z'')}} can be computed by a change of variables from ''U'',''V'' to ''Y'',''Z'', and {{math|''Y''}} can be derived by marginalizing out {{math|''Z''}} from the joint density.

The inverse transformation is
<math display="block">\begin{align}
U &= YZ \\
V &= Z
\end{align}</math>

The absolute value of the Jacobian matrix determinant <math>J(U,V\mid Y,Z)</math> of this transformation is:
<math display="block">
\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| .
</math>

Thus:
<math display="block">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| .</math>

And the distribution of {{math|''Y''}} can be computed by marginalizing out {{math|''Z''}}:
<math display="block">p(y) = \int_{-\infty}^\infty p_U(yz)\,p_V(z)\, |z| \, dz</math>

This method crucially requires that the transformation from ''U'',''V'' to ''Y'',''Z'' be bijective.  The above transformation meets this because {{math|''Z''}} can be mapped directly back to {{math|''V''}}, and for a given {{math|''V''}} the quotient {{math|''U''/''V''}} is monotonic.  This is similarly the case for the sum {{math|''U'' + ''V''}}, difference {{math|''U'' − ''V''}} and product {{math|''UV''}}.

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

===Example: Quotient of two standard normals===
Given two standard normal variables {{math|''U''}} and {{math|''V''}}, the quotient can be computed as follows.  First, the variables have the following density functions:
<math display="block">\begin{align}
p(u) &= \frac{1}{\sqrt{2\pi}} e^{-{u^2}/{2}} \\[1ex]
p(v) &= \frac{1}{\sqrt{2\pi}} e^{-{v^2}/{2}}
\end{align}</math>

We transform as described above:
<math display="block">\begin{align}
Y &= U/V \\[1ex]
Z &= V
\end{align}</math>

This leads to:
<math display="block">\begin{align}
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{align}</math>

This is the density of a standard [[Cauchy distribution]].

==See also==
* {{Annotated link|Density estimation}}
* {{Annotated link|Kernel density estimation}}
* {{Annotated link|Likelihood function}}
* {{Annotated link|List of probability distributions}}
* {{Annotated link|Probability amplitude}}
* {{Annotated link|Probability mass function}}
* {{Annotated link|Secondary measure}}
* Uses as ''position probability density'':
** {{Annotated link|Atomic orbital}}
** {{Annotated link|Home range}}

==References==
{{reflist}}

==Further reading==
* {{cite book
 | first = Patrick |last=Billingsley
 | title = Probability and Measure
 | publisher = John Wiley and Sons
 | location = New York, Toronto, London
 | year = 1979
 | isbn = 0-471-00710-2}}
* {{cite book |first1=George |last1=Casella |first2=Roger L. |last2=Berger |author-link2=Roger Lee Berger |title=Statistical Inference |edition=Second |publisher=Thomson Learning |year=2002 |isbn=0-534-24312-6 |pages=34–37 }}
* {{cite book
 | first = David |last=Stirzaker
 | year = 2003
 | title = Elementary Probability
 |publisher=Cambridge University Press
 | isbn = 0-521-42028-8
 | url-access = registration
 | url = https://archive.org/details/elementaryprobab0000stir
 }} Chapters 7 to 9 are about continuous variables.

==External links==
* {{Springer
 |title=Density of a probability distribution
 |id=D/d031110
 |first=N.G. |last=Ushakov
}}
* {{MathWorld|ProbabilityDensityFunction}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Probability Density Function}}
[[Category:Functions related to probability distributions]]
[[Category:Equations of physics]]

{{Sourceattribution|Probability density function}}