2.10 ( ) Using the property (x + 1) = x(x) of the gamma function, derive the...

2.10 ( ) Using the property Γ(x + 1) = xΓ(x) of the gamma function, derive the following results for the mean, variance, and covariance of the Dirichlet distribution given by (2.38)

E[μj] = αj

α0

(2.273)

var[μj] = αj(α0 − αj)

α2 0(α0 + 1)

(2.274)

cov[μjμl] = − αjαl

α2 0(α0 + 1), j = l (2.275)

where α0 is defined by (2.39).