Here's my problem:

Suppose we have a three-stage hierarchical model

Y|N~ Binomial(N,p), N|λ ~ Poisson(λ), λ ~ Gamma(a,b)

Find a marginal distribution for Y.

What I did was first find the pdf for f(N,λ) by multiplying f(N|λ) * f(λ).

Next step was to find a marginal probability for N. I integrated f(N,λ) with respect to λ from 0 to infinity which worked out somewhat nicely giving me the marginal distribution for N.

here's what I got: Γ(N+a) ((b/(b+1))^(N+a)/(N!b^aΓ(a)),

or, in a more easily looked at format, here

Now, I found f(N,Y) which is just f(N)*f(Y|N).

The problem arises when I try to obtain the marginal for Y. I've been looking for a pdf to help make this easier, but I can't see any that do any good.

Thanks for you help!