= Geometric count under Bernoulli thinning
{title2=$p_\alpha=p/(p+(1-p)\alpha)$}
Independently retain each event of a zero-based <geometric distribution> count with probability $\alpha$. Conditional on the original count, the retained count is binomial, so its <probability generating function> is $p/[1-(1-p)(1-\alpha+\alpha z)]$. This is again a zero-based <geometric distribution>, with success parameter $p_\alpha$ and mean $\alpha(1-p)/p$. In <excess of loss reinsurance>, $\alpha$ is the probability that a claim exceeds the retention.
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