Derive the density for the beta and Dirichlet, given by XXXXXXXXXXand XXXXXXXXXX), arising from the characterization of a gamma random variable U divided by a sum of gammas. The characterization of...

Derive the density for the beta and Dirichlet, given by (11.3.14) and (11.3.15), arising from the characterization of a gamma random variable U divided by a sum of gammas.

The characterization of the noncentral chi-square from Q = Z2 i , where Zi ∼ normal(µi, σ2), suggests another route. Derive an algorithm and compare its performance with the Poisson mixed chi-square approach suggested in the text.