Probability density function

Boxplot and probability density function (pdf) of a gaussian probability distribution N(0, σ²)

In probability theory, a probability density function (abbreviated as pdf, or just density) of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point in the observation space. The probability of a random variable falling within a given set is given by the integral of its density over the set.

The terms “probability distribution function”^[1] and "probability function"^[2] have also been used to denote the probability density function. However, special care should be taken around this usage since it is not standard among probabilists and statisticians. In other sources, “probability distribution function” may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density.

1 Continuous univariate distributions
2 Formal definition
- 2.1 Discussion
3 Further details
4 Link between discrete and continuous distributions
5 Probability functions associated with multiple variables
6 Sums of independent random variables
7 Dependent variables and change of variables
- 7.1 Multiple variables
8 Finding moments and variance
9 See also
10 References
11 Bibliography
12 External links

Continuous univariate distributions

A probability density function is most commonly associated with continuous univariate distributions. A random variable X has density ƒ, where ƒ is a non-negative Lebesgue-integrable function, if:

$\operatorname P [a \leq X \leq b] = \int_a^b f(x) \, \mathrm{d}x .$

Hence, if F is the cumulative distribution function of X, then:

$F(x) = \int_{-\infty}^x f(u) \, \mathrm{d}u ,$

and (if ƒ is continuous at x)

$f(x) = \frac{\mathrm{d}}{\mathrm{d}x} F(x) .$

Intuitively, one can think of ƒ(x) dx as being the probability of X falling within the infinitesimal interval [x, x + dx].

Formal definition

This definition may be extended to any probability distribution using the measure-theoretic definition of probability. A random variable X has probability distribution X_*P: the density of X with respect to a reference measure μ is the Radon–Nikodym derivative:

$f = \frac{\mathrm d X_*\operatorname{P}}{\mathrm d \mu} .$

That is, ƒ is any function with the property that:

$\mathrm P [X \in A ] = \int_{X^{-1}A} \, \mathrm d \operatorname{P} = \int_A f \, \mathrm d \mu$

for any measurable set A.

Discussion

In the continuous univariate case above, the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space (usually the set of integers, or some subset thereof).

Note that it is not possible to define a density with reference to an arbitrary measure (i.e. one can't choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost everywhere unique.

Further details

For example, the continuous uniform distribution on the interval [0, 1] has probability density ƒ(x) = 1 for 0 ≤ x ≤ 1 and ƒ(x) = 0 elsewhere.

The standard normal distribution has probability density

$f(x)={e^{-{x^2/2}}\over \sqrt{2\pi}}.$

If a random variable X is given and its distribution admits a probability density function ƒ, then the expected value of X (if it exists) can be calculated as

$\operatorname{E}(X)=\int_{-\infty}^\infty x\,f(x)\,dx.$

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 F(x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its derivative can be used as probability density:

$\frac{d}{dx}F(x) = f(x).$

If a probability distribution admits a density, then the probability of every one-point set {a} is given by the probability distribution.

Two probability densities ƒ and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

In the field of statistical physics, 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 ƒ(t) dt, or:

$\Pr(t<X<t+dt) = f(t)\,dt.~$

Link between discrete and continuous distributions

The definition of a probability density function at the start of this page makes it possible to describe the variable associated with a continuous distribution using a set of binary discrete variables associated with the intervals [a; b] (for example, a variable being worth 1 if X is in [a; b], and 0 if not).

It is also possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. For example, let us consider a binary discrete random variable taking −1 or 1 for values, with probability ½ each.

The density of probability associated with this variable is:

$f(t) = \frac{1}{2}(\delta(t+1)+\delta(t-1)).$

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

$f(t) = \sum_{i=1}^nP_i\, \delta(t-x_i),$

where $x_1, \ldots, x_n$ are the discrete values accessible to the variable and $P_1, \ldots, P_n$ are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution.

Probability functions associated with multiple variables

For continuous random variables $X_1,\ldots,X_n$ , 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 $X_1,\ldots,X_n$ , the probability that a realisation of the set variables falls inside the domain D is

$\Pr \left( X_1,\ldots,X_N \isin D \right) = \int_D f_{X_1,\dots,X_n}(x_1,\ldots,x_N)\,dx_1 \cdots dx_N.$

If F(x₁, …, x_n) = Pr(X₁ ≤ x₁, …, X_n ≤ x_n) is the cumulative distribution function of the vector (X₁, …, X_n), then the joint probability density function can be computed as a partial derivative

$f(x) = \frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \bigg|_x$

Marginal densities

For i=1, 2, …,n, let $f_{X_i}(x_i)$ be the probability density function associated to variable $X_i$ alone. This is called the “marginal” density function, and can be deduced from the probability densities associated of the random variables $X_1,\ldots,X_n$ by integrating on all values of the n − 1 other variables:

$f_{X_i}(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_{i-1}\,dx_{i+1}\cdots dx_n .$

Independence

Continuous random variables $X_1,\ldots,X_n$ are all independent from each other if and only if

$f_{X_1,\dots,X_n}(x_1,\ldots,x_n) = f_{X_1}(x_1)\cdots f_{X_n}(x_n).$

Corollary

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

$f_{X_1,\dots,X_n}(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n),$

then the n variables in the set are all independent from each other, and the marginal probability density function of each of them is given by

$f_{X_i}(x_i) = \frac{f_i(x_i)}{\int f_i(x)\,dx}.$

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 $\vec R$ a 2-dimensional random vector of coordinates $(X,Y)$ : the probability to obtain $\vec R$ in the quarter plane of positive x and y is

$\Pr \left( X > 0, Y > 0 \right) = \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.$

Sums of independent random variables

The probability density function of the sum of two independent random variables U and V, each of which has a probability density function, is the convolution of their separate density functions:

$f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy = \left( f_{U} * f_{V} \right) (x)$

It is possible to generalize the previous relation to a sum of N independent random variables, with densities $U_{1}, \ldots, U_{N}$ :

$f_{U_{1} + \ldots + U_{N}}(x) = \left( f_{U_{1}} * \ldots * f_{U_{N}} \right) (x)$

Dependent variables and change of variables

If the probability density function of an independent random variable $X$ is given as $f_X(x)$ , 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 $f_{g(X)}=f_Y$ using a known (for instance uniform) random number generator.

If the function g is monotonic, then the resulting density function is

$f_Y(y) = \left| \frac{1}{g'(g^{-1}(y))} \right| \cdot f_X(g^{-1}(y)).$

Here $g^{-1}$ denotes the inverse function and g' denotes the derivative.

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

$\left| f_Y(y)\, dy\right| = \left| f_X(x)\, dx\right|,$

$f_Y(y) = \left| \frac{dx}{dy} \right| f_X(x) = \left| \frac{1}{g'(x)} \right| f_X(x) = \left| \frac{1}{g'(g^{-1}(y))} \right|f_X(g^{-1}(y)).$

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

$\sum_{k}^{n(y)} \left| \frac{1}{g'(g^{-1}_{k}(y))} \right| \cdot f_X(g^{-1}_{k}(y))$

where $n(y)$ is the number of solutions in x for the equation $g(x) = y$ , and $g^{-1}_{k}(y)$ are these solutions.

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

$E(g(X)) = \int_{-\infty}^\infty y f_{g(X)}(y)\,dy,$

one may find instead

$E(g(X)) = \int_{-\infty}^\infty g(x) f_X(x)\,dx.$

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 one-to-one function. In some cases the latter integral is computed much more easily than the former.

Multiple variables

The above formulas can be generalized to variables (which we will again call y) depending on more than one other variable. $f(x_0, x_1, \dots, x_{m-1})$ shall denote the probability density function of the variables y depends on, and the dependence shall be $y=g(x_0, x_1, \dots, x_{m-1})$ . Then, the resulting density function is

$\int\limits_{y = g(x_0, x_1, \dots, x_{m-1})} \frac{f(x_0, x_1,\dots, x_{m-1})}\sqrt{\sum_{j=0}^{j<m} (\frac{\partial g}{\partial x_j}(x_0, x_1, \dots , x_{m-1}))^2} \; dV$

where the integral is over the entire (m-1)-dimensional solution of the subscripted equation and the symbolic dV must be replaced by a parametrization of this solution for a particular calculation; the variables $x_0, x_1, \dots, x_{m-1}$ are then of course functions of this parametrization.

This derives from the following, perhaps more intuitive representation: Suppose x is an n-dimensional random variable with joint density f. If $\mathbf{y} = H(\mathbf{x})$ , where H is a bijective, differentiable function, then y has density g:

$g(\mathbf{y}) = f(\mathbf{x})\left\vert \det\left(\frac{d\mathbf{x}}{d\mathbf{y}}\right)\right \vert$

with the differential regarded as the Jacobian of the inverse of H, evaluated at y.

Finding moments and variance

In particular, the nth moment $E(X^n)$ of the probability distribution of a random variable X is given by

$E(X^n) = \int_{-\infty}^\infty x^n f_X(x)\,dx,$

and the variance is

$\operatorname{var}(X) = E[(X - E(X))^2] = \int_{-\infty}^\infty (x-E(X))^2 f_X(x)\,dx$

$\operatorname{var}(X) = E(X^2) - [E(X)]^2.\,$

References

↑ [1] PlanetMath
↑ [2] Mathworld

Ushakov, N.G. (2001), "Density of a probability distribution", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/D/d031110.htm

Bibliography

Pierre Simon de Laplace (1812). Analytical Theory of Probability.

The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.

Andrei Nikolajevich Kolmogorov (1950). Foundations of the Theory of Probability.

The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.

Patrick Billingsley (1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons.

David Stirzaker (2003). Elementary Probability.

Chapters 7 to 9 are about continuous variables. This book is filled with theory and mathematical proofs.

External links

Weisstein, Eric W., "Probability density function" from MathWorld.

Theory of probability distributions

probability mass function (pmf) · probability density function (pdf) · cumulative distribution function (cdf) · quantile function

raw moment · central moment · mean · variance · standard deviation · skewness · kurtosis · L-moment

moment-generating function (mgf) · characteristic function · probability-generating function (pgf) · cumulant