= Finite Erlang-mixture tail for a negative-binomial exponential aggregate
{title2=$\mathbb P(S>x)=\sum_{n=1}^k\binom{k}{n}(1-p)^np^{k-n}e^{-p\lambda x}\sum_{j=0}^{n-1}(p\lambda x)^j/j!$}
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 $\operatorname{Bin}(k,1-p)$ 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.
Back to article page
