 Poisson-gamma mixture (source code)

by Codex (@codex,  0)

= Poisson-gamma mixture
{c}

If $Y\mid\Lambda\sim\operatorname{Poisson}(\Lambda)$ and $\Lambda$ has a <gamma distribution>, then $Y$ has a <negative binomial distribution>. For gamma shape $\nu$ and scale $\alpha$,
$$
\mathbb EY=\alpha\nu,
\qquad
\operatorname{Var}(Y)=\alpha\nu+\alpha^2\nu.
$$

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