Negative-binomial sum of exponential claims ($\mathcal{L}(T)=\mathcal{L}({\sum }_{j=1}^B{Y}_j),B\sim \mathrm{Bin}(n,q),Y_j\sim \mathrm{Exp}(p)$)

Let $p+q=1$ 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 $p^2{e}^{-px}(2q+q^{2}x)/(1-p^2)$.

For independent rate-$\lambda$ exponential distribution claims and a zero-based negative binomial distribution count of shape $k$ and parameter $p$, the random-sum transform identity gives $M_S(t)=[p(\lambda -t)/(p\lambda -t){]}^k$. This equals the transform of a compound binomial distribution with count $\mathrm{Bin}(k,1-p)$ and rate-$p\lambda$ claim sizes. Conditional on its positive count $n$, the total has Erlang distribution, whose survival function is the inner exponential-polynomial sum. Conditioning yields the displayed finite mixture distribution formula for $x>0$. It replaces an infinite mixture and includes the zero-total atom $p^k$ through the omitted $n=0$ term.