Finite Erlang-mixture tail for a negative-binomial exponential aggregate ($\mathbb{P}(S>x)={\sum }_{n=1}^k\left(\genfrac{}{}{0ex}{}{k}{n}\right)(1-p{)}^n{p}^{k-n}{e}^{-p\lambda x}{\sum }_{j=0}^{n-1}(p\lambda x{)}^j/j!$)

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.