Definitions

# Convolution sampling

In mathematics, convolution sampling is a technique used to generate observations 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 $X sim Erlang\left(k, theta\right)$, defined as the sum of k random variables each with distribution $exp\left(k theta\right)$.

Notice that:

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

One can now generate $Erlang\left(k, theta\right)$ samples using the sampler for the exponential distribution:

if $X_i sim exp\left(k theta\right)$ then $X=sum_\left\{i=1\right\}^k X_i sim Erlang\left(k,theta\right)$

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