Convolution random number generator

In mathematics, convolution sampling is a technique used to generate random variates from a distribution.

A number of distributions can be expressed in terms of the (possibly weighted) sum of two or more random variables from other distributions (The distribution of the sum is the convolution of the distributions of the individual random variables).

Example

Consider the random variable $X\ \sim Erlang(k,\theta )$, defined as the sum of k random variables each with an exponential distribution $Exp(k\theta )\,$:see Gamma distribution.

Notice that:

$E[X]={\frac {1}{k\theta }}+{\frac {1}{k\theta }}+...+{\frac {1}{k\theta }}={\frac {1}{\theta }}.$

One can now generate $Erlang(k,\theta )$ samples using a random number generator for the exponential distribution:

if $X_{i}\ \sim exp(k\theta )$ then $X=\sum _{i=1}^{k}X_{i}\sim Erlang(k,\theta ).$