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!