I'm trying to do point (b) of exercise 3.30 of the book "Statistical Inference" (Casella & Berger).
The exercise says:
"Use the identities of Theorem 3.4.2. to
(a) calculate the variance of a binomial random variable.
(b) calculate the mean and variance of a beta(a,b) random variable."
Theorem 3.4.2 says the following.
If X is a random variable with pdf or pmf of the form
(exponential family), then
Point (a) is doable, but what about point (b)? First of all, since we have two parameters (a and b), do we get a system of two equations? My difficulties is that I have a cumbersome E[logX] and then I have to take derivatives of B(a,b). It's a nightmare... If I'm not wrong, .
and there are lots of calculations for and other related terms. And as you expected you will need to in terms of some non-elementary function like the digamma function which is the derivative of log-gamma function.