In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 and computer software

Computer software

Computer software, or just software, is a collection of computer programs and related data that provide the instructions for telling a computer what to do and how to do it....

, a convolution random number generator is a pseudo-random number sampling

Pseudo-random number sampling

Pseudo-random number sampling or non-uniform pseudo-random variate generation is the numerical practice of generating pseudo-random numbers that are distributed according to a given probability distribution....

 method that can be used to generate random variate

Random variate

A random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable would have different values. Random variates are used when simulating processes driven by random influences...

s from certain classes of probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

. The particular advantage of this type of approach is that it allows advantage to be taken of existing software for generating random variates from other, usually non-uniform, distributions. However, faster algorithms may be obtainable for the same distributions by other more complicated approaches.

A number of distributions can be expressed in terms of the (possibly weighted) sum of two or more random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s from other distributions. (The distribution of the sum is the convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 of the distributions of the individual random variables).

Example

Consider the problem of generating a random variable with an Erlang distribution, . Such a random variable can be defined as the sum of k random variables each with an exponential distribution

Exponential distribution

In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 . This problem is equivalent to generating a random number for a special case of the Gamma distribution, in which the shape parameter

Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...

takes an integer value.

Notice that:


One can now generate samples using a random number generator for the exponential distribution:

if     then