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Negative binomial distribution

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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 $\alpha$,
$$
\mathbb EY=\alpha\nu,
\qquad
\operatorname{Var}(Y)=\alpha\nu+\alpha^2\nu.
$$

Poisson-gamma mixture

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 218 / 1 / e / i / Solution

Poisson-gamma mixture

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Poisson-gamma mixture

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