= Categorical thinning of a Poisson claim count
{title2=$N_i\text{ independent},\quad N_i\sim\operatorname{Pois}(\lambda p_i)$}
Assign each of $N\sim\operatorname{Pois}(\lambda)$ claims with <independent> category labels to one of finitely many categories with <probabilities> $p_i$. Conditional counts have a <multinomial distribution> with <probability generating function> $(\sum_i p_i z_i)^N$. Averaging over $N$ gives $\exp(\lambda(\sum_i p_i z_i-1))=\prod_i\exp(\lambda p_i(z_i-1))$. This proves <independent> category counts and supports discrete-severity <compound Poisson distribution> representations.
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