(Advanced) The gamma distribution can be defined as f(|,) = 1 ( 1)! 1 exp(/) (5.40) for 0, a positive

(Advanced) The gamma distribution can be defined as f(λ|α,β) = 1

(α −1)!β

α

λ

α−1 exp(−λ/β) (5.40)

for λ ≥ 0, α a positive integer, and β > 0. (α is not required to be an integer, but we will assume this for simplicity here.)

Assuming the hierarchical distribution for y given by Equations (5.5) and (5.6), show that the marginal distribution of y is negative binomial with parameters r and p.