 Negative-binomial sum of exponential claims

Let $n$ be a positive integer, let $0<p<1$, and put $q=1-p$ and let the original independent count have negative binomial distribution with probability mass function $\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)p^n{q}^k$ on $k\ge 0$. With unit-rate exponential distribution severities, its moment-generating function is $[p+qp/(p-t){]}^n$ for $t<p$. Thus it also has a random sum of independent claims representation with independent binomial distribution count $B$ and rate-$p$ exponential distribution steps. For $n=2$ the law is $p^2{\delta }_0+2pq\mathrm{Exp}(p)+q^2\mathrm{Erlang}(2,p)$. The hurdle decomposition of a positive random sum therefore has zero mass $p^2$ and positive density $p^2{e}^{-px}(2q+q^{2}x)/(1-p^2)$.

The positive-integer restriction on $n$ is essential for the finite binomial distribution representation. A negative binomial distribution also permits noninteger positive shape through its generalized probability mass function, but that broader model cannot be represented by a binomial distribution with a noninteger number of trials.