# An alternative to Beta-Binomial distribution?

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).

## 1 Answer

Your distribution seems to be a binomial distribution with parameters $$n$$ and $$\frac{\alpha}{\alpha+\beta}$$. This has support on $$\{0,1,2,\ldots,n\}$$ and mean $$\frac{n\alpha}{\alpha+\beta}$$ and variance $$\frac{n\alpha\beta}{(\alpha+\beta)^2}$$

The original Beta-binomial distribution also has support on $$\{0,1,2,\ldots,n\}$$ and mean $$\frac{n\alpha}{\alpha+\beta}$$, but its variance is $$\frac{n\alpha\beta}{(\alpha+\beta)^2} \frac{\alpha+\beta+n}{\alpha+\beta+1}$$, which is rather larger

Essentially, more extreme values near $$0$$ and $$n$$ are more likely in the original Beta-binomial distribution than in your distribution