Nested Poisson flood count ($G_N(z)=\exp \{\nu (e^{\lambda (z-1)}-1)\}$)

If a Poisson distribution count $M$ of clusters has mean $\nu$, and each cluster independently contains a Poisson number of claims of mean $\lambda$, then the total count is a compound Poisson distribution with the displayed probability generating function. Conditioning gives $\mathbb{E}N=\nu \lambda$ and $\mathrm{Var}N=\nu \lambda (1+\lambda )$, so it is not Poisson for positive parameters. With independent severities of mean $\mu$ and variance ${\sigma }^2$, the aggregate payment has mean $\nu \lambda \mu$ and variance $\nu [\lambda ({\sigma }^2+{\mu }^2)+{\lambda }^2{\mu }^2]$. Independent validity thinning replaces the per-cluster count mean by $\lambda (1-p)$ and multiplies expected payment by $1-p$.