Erlang distribution
= Erlang distribution
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The sum of $n$ independent exponential variables of rate $r$ has the Erlang distribution with density
$$
f_n(x)=\frac{r^nx^{n-1}}{(n-1)!}e^{-rx},
\qquad x>0.
$$
Convolving with one more exponential proves the formula inductively.