I'm reading about Beta-Binomial. Assume a r.v. x~Bin(n, pi). (pi) in itself, is a r.v. drawn from Beta(alpha, beta) I was thinking, may be I can take the Expected value of pi (which is alpha/(alpha+beta) ) and substitute it in the original Binomial distribution and I'm done, I got P(x). My question is: what is the difference between the distribution that I got, and Beta-Binomial?
PS: Beta-Binomial finds the joint distribution then marginalizes (pi) to find the predictive distribution P(x).