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For the hierarchical model Y |Λ ∼ Poisson(Λ) and Λ ∼ Gamma(α, β), find the marginal...

For the hierarchical model Y |Λ ∼ Poisson(Λ) and Λ ∼ Gamma(α, β), find the marginal distribution, mean, and variance of Y . Show that the marginal distribution of Y is a negative binomial if α is an integer. (b) Show that the three-stage model Y|N∼Binomial(N,p), N|Λ∼Poisson(Λ), andΛ∼Gamma(α,β) leads to the same marginal distribution of Y .

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