= Gamma-mixed Poisson aggregate with exponential claims
{title2=$p^2\delta_0+2pq\,\operatorname{Exp}(p/\mu)+q^2\operatorname{Gamma}(2,p/\mu)$}
If $\Lambda$ has <gamma distribution> with shape $2$ and rate $p/q$, $N\mid\Lambda$ has <Poisson distribution> with intensity $\Lambda$, and the independent claim sizes have <exponential distribution> of mean $\mu$, where $p+q=1$, the aggregate law is $p^2\delta_0+2pq\,\operatorname{Exp}(p/\mu)+q^2\operatorname{Gamma}(2,\text{rate }p/\mu)$. This follows by expanding its <moment-generating function> as $[p+q\,p/(p-\mu t)]^2$.
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