= Binomial accident portfolio with Poisson clusters
{title2=$G_S(z)=(1-p+pe^{\lambda(z-1)})^m$}
For independent factories with accident probability $p$, and independent <Poisson distribution> claim counts of mean $\lambda$ per accident, the accident count is binomial and each accident supplies a cluster. The <random-sum transform identity> gives the displayed <probability generating function>. In particular the no-claim chance is $(1-p+pe^{-\lambda})^m$; a factory with an accident can still have zero claims. The binomial accident law has Panjer parameters $a=-p/(1-p)$ and $b=(m+1)p/(1-p)$.
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