One thing I forgot to mention when talking about the beta distribution and its use in Bayesian analysis is the normalizing constant. The beta distribution with shape parameters a and b and variable p is:
p**(a-1) * (1-p)**(b-1)
where the range of the function is from 0 to 1. To be a probability density function, of course, the total density over this range must equal 1.
So, we need to find a constant k which normalizes the beta distribution function. If you think about it for a minute, it should be obvious that k equals the inverse of the integral of the pdf over its range. It turns out that k can be expressed in terms of the gamma function:
The gamma function is a complex pdf in its own right. But a simple fact to remember about it is that if a is a positive integer (as it will be for our beta distributions), then
The fact that k is expressed in terms of the gamma function is important for the derivation of expressions for the mean and variance and updating functions of the beta distribution. But we don't need to think about them to use the beta distribution for analysis of binomial proportions.