= Compound mixed Poisson distribution
{title2=$S=\sum_{j=1}^{N}X_j,\quad N\mid\Lambda\sim\operatorname{Poisson}(\Lambda)$}
A compound mixed Poisson distribution is a <random sum of independent claims> whose count has a <Poisson mixture> law: conditional on a nonnegative random intensity $\Lambda$, $N$ has <Poisson distribution> with parameter $\Lambda$. The claim sizes are <independent and identically distributed random variables>, independent of the intensity and count mechanism. If $M_X$ is their <moment-generating function>, then $M_S(t)=\mathbb E\exp(\Lambda(M_X(t)-1))$ wherever finite. A <gamma distribution> intensity of shape $r$ and rate $\alpha$ gives a <negative binomial distribution> count with shape $r$ and success probability $\alpha/(\alpha+1)$. Independent intensities with common gamma rate add their shapes by <additivity of independent gamma distributions with a common rate>.
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