 Finite Erlang-mixture tail for a negative-binomial exponential aggregate

Let $k$ be a positive integer, $0<p<1$, and $\lambda >0$. 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.

For $k=1$, the formula reduces to $(1-p)e^{-p\lambda x}$ for $x>0$: there is an atom $p$ at zero and a rate-$p\lambda$ exponential distribution positive component. This checks both the omitted zero-count term and the parameter scaling.