Can someone please walk me through how to find the mean and variance of the beta distribution (with parameters alpha and beta)? Also, I have seen two pdfs for the beta distribution. One of them varies from 0 to infinite, while the other varies from 0 to 1. I was wondering what the difference between these two pdfs are?
Once you know that the normalizing factor of the density of the beta distribution with parameters $(a,b)$ is $1/B(a,b)$, you know without calculus that the moments of a random variable $X$ with this distribution are $\mathrm E(X^s)=B(a+s,b)/B(a,b)$ and, more generally, $\mathrm E(X^s(1-X)^t)=B(a+s,b+t)/B(a,b)$.
The rest is here.