Probability density function

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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f is a non-negative Lebesgue-integrable function $\mathbb{R}\to\mathbb{R}$ such that the probability of the interval [a, b] is given by

$\int_a^b f(x)\,dx$

for any two numbers a and b. This implies that the total integral of f must be 1. Conversely, for any non-negative Lebesgue-integrable function f with total integral 1, there must be some probability distribution for which f represents the probability density.

Intuitively, if a probability distribution has density f(x), then the infinitesimal interval [x, x + dx] has probability f(x) dx.

Informally, a probability density function can be seen as a "smoothed out" version of a histogram: if one empirically samples enough values of a continuous random variable, producing a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density, assuming that the output ranges are sufficiently narrow.

Any function f that describes the probability density in terms of the input variable x in a manner described below is a probability density function.

$f(x) \geq 0$ for all $x$
The total area under the graph is 1:

$\int_{-\infty}^\infty \,f(x)\,dx = 1.$

The actual probability can then be calculated by taking the integral of the function f(x) by the integration interval of the input variable x.

For example: the probability of the variable X being within the interval [4.3,7.8] would be

$\Pr(4.3 \leq X \leq 7.8) = \int_{4.3}^{7.8} f(x)\,dx.$

1 Further details
2 Link between discrete and continuous distributions
3 Probability function associated to multiple variables
4 Sums of independent random variables
5 Dependent variables and change of variables
- 5.1 Multiple variables
6 Finding moments and variance
7 Bibliography
8 See also

Further details

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

It is a common mistake to think of f(x) as the probability of {x}, but this is incorrect; in fact, f(x) will often be bigger than 1 - consider a random variable that is uniformly distributed between 0 and ½. Loosely, one may think of f(x) dx as the probability that a random variable whose probability density function is f , is in the interval from x to x + dx, where dx is an infinitely small increment.

Two probability densities f 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 $f(t)\,dt$ , 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 using a density of probability, via 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 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.

In physics, this description is also useful in order to characterize mathematically the initial configuration of a Brownian movement.

Probability function associated to 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

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

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:

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

Dependent variables and change of variables

If the probability density function of an independent random variable x is given as f(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable y which depends on x. This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape f using a known (for instance uniform) random number generator. If the dependence is y = g(x) and the function g is monotonic, then the resulting density function is

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

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,

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

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

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

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

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 of g(X). However, rather than computing

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

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 variables. f(x₀, x₁, ..., x_m-1) shall denote the probability density function of the variables y depends on, and the dependence shall be y = g(x₀, x₁, ..., x_m-1). Then, the resulting density function is

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

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 x₀, x₁, ..., 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 y = H(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(Xⁿ) 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

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

or, expanding, gives:

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

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 books is filled with theory and mathematical proofs.