Probability density function

In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. A probability density function is non-negative everywhere and its integral from &minus;&infin; to +&infin; is equal to 1. If a probability distribution has density f(x), then intuitively 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 measures values of a continuous random variable repeatedly and produces a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density (assuming that the variable is sampled sufficiently often and the output ranges are sufficiently narrow).

Formally, a probability distribution has density f(x) if f(x) is a non-negative Lebesgue-integrable function R &rarr; 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, any non-negative Lebesgue-integrable function with total integral 1 is the probability density of a suitably defined probability distribution.

Simplified explanation
A probability density function is any function f(x) that describes the probability density in terms of the input variable x in a manner described below.
 * f(x) is greater than or equal to zero for all values of 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 variable x being within the interval [4.3,7.8] would have the actual probability of


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

Further details
For example, the continuous uniform distribution on the interval [0,1] has probability density f(x) = 1 for 0 &le; x &le; 1 and zero 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(a) as the probability of {a}, but this is incorrect; in fact, f(a) will often be bigger than 1 - consider a random variable with a uniform distribution between 0 and 1/2.

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 &minus;1 or 1 for values, with for probability 1/2 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:


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

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. 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 \ldots dx_N.$$

For i in [1, n], let $$f_{X_i}(x_i)$$ the probability density function associated to variable $$X_i$$ alone. This probability density 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 &minus; 1 other variables:


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

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


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

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 x is


 * $$\Pr \left( X > 0, Y > 0 \right)

= \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.$$