Convolution random number generator

Consider the problem of generating a random variable with an Erlang distribution, ${\displaystyle X\ \sim \operatorname {Erlang} (k,\theta )}$. Such a random variable can be defined as the sum of k random variables each with an exponential distribution ${\displaystyle \operatorname {Exp} (k\theta )\,}$. This problem is equivalent to generating a random number for a special case of the Gamma distribution, in which the shape parameter takes an integer value.

Notice that:

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

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

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