Method of Moments Estimation

Probably the most logical (and certainly the most straight-forward) approach to determining the parameters for the Beta distribution uses the moments-specifically the sample mean and variance-to solve for the parameters a and b. Recall that a random variable $p \sim \rm {Beta} (a,b)$ has mean and variance given by

$$\bar p = {a \over {a+b}}$$ (15)

$$\sigma_p^2 = {{a b} \over {(a+b+1)(a+b)^2}}$$ (16)

These equations can be solved to find explicit solutions for a and b in terms of E(p) and Var(p).

$$a = \left [ {{\bar p(1-\bar p)} \over {\sigma_p^2}} - 1 \right ] \bar p$$ (17)

$$b = \left [ {{\bar p(1-\bar p)} \over {\sigma_p^2}} - 1 \right ] (1-\bar p)$$ (18)