2.13 Suppose that the random variable X Po() and that (, ), where is a positive integer. Show that the marginal

2.13 Suppose that the random variable X ∼ Po(λ) and that λ ∼ Γ(α, β), where α is a positive integer. Show that the marginal distribution of X has the negative-binomial form:

pr(X = k) = 

k + α − 1

α − 1

 1

(1 + β)α

βk

(1 + β)k , k ≥ 0 .