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)

so (when it exists) is simply the derivative of the distribution function

(3)

Similarly, the distribution function is related to a discrete probability by

			(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)

The probability content of a closed region can be found much more efficiently than by direct integration of the probability density function by appropriate evaluation of the distribution function at all possible extrema defined on the region (Rose and Smith 1996; 2002, p. 193). For example, for a bivariate distribution function , the probability content in the region , is given by

(10)

but can be computed much more efficiently 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. The method of finding the value at which the distribution is a maximum is known as the maximum likelihood method.