Distribution Function

The distribution function , also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate takes on a value less than or equal to a number . The distribution function is sometimes also denoted (Evans et al. 2000, p. 6).

The distribution function is therefore related to a continuous probability density function by

			(1)
			(2)

where integration may be restricted to the support of the probability density function, since outside that support. Thus (when it exists) is the derivative of the distribution function

(3)

For a discrete distribution with probability mass function , the distribution function is

			(4)
			(5)

There exist distributions that are neither continuous nor discrete.

A joint distribution function can be defined if outcomes are dependent on two parameters:

			(6)
			(7)
			(8)

Similarly, a multivariate distribution function can be defined if outcomes depend on parameters:

(9)

For a bivariate distribution with joint probability density function and distribution function , the probability of a rectangle can be evaluated using the distribution function (Rose and Smith 1996; Rose and Smith 2002, p. 193). The probability in the region , is given by

(10)

and can also be computed using

(11)

Given a continuous , assume you wish to generate numbers distributed as using a random number generator. If the random number generator yields a uniformly distributed value in for each trial , then compute

(12)

The formula connecting with a variable distributed as is then

(13)

where is the inverse function of . For example, if were a normal distribution so that

(14)

then

(15)

A distribution with constant variance of for all values of is known as a homoscedastic distribution. A value at which the probability density function attains its maximum is a mode.

Cumulative distribution functions are computed in the Wolfram Language using CDF.