The Beta distribution is defined by f(X) = ( + ) ()()X1(1 X)1 for 0 < X < 1 = 0 elsewhere where

The Beta distribution is defined by f(X) =

Γ(α + β)

Γ(α)Γ(β)Xα−1(1 − X)β−1 for 0< X < 1

= 0 elsewhere where α > 0 and β > 0. This is a skewed continuous distribution.

(a) For α = β = 1 this reverts back to the Uniform (0, 1) probability density function. Show that E(X) = (α/α + β) and var(X) = αβ/(α + β)2(α + β + 1).

(b) Suppose that α = 1, find the estimators of β using the method of moments and the method of maximum likelihood.