The Dirichlet density and the density of a transformation of the Dirichlet distribution makes good test problems for integration in many dimensions. The density function for the Dirichlet is          ...

The Dirichlet density and the density of a transformation of the Dirichlet distribution makes good test problems for integration in many dimensions. The density function for the Dirichlet is

                            f (t) = ( d+1 j=1 αj ) d+1 j=1 (αj ) d j=1 t αj −1 j (1 − d j=1 tj ) αd+1−1

with support only on the simplex 0 ≤ tj ≤ 1, tj ≤ 1. The transformation extends the support to Rd with the density

                            g(x) = ( d+1 j=1 αj ) d+1 j=1 (αj ) e d j=1 αj xj (1 + d j=1 exj ) d+1 j=1 αj .

a) Verify the calculus for the transformed Dirichlet density g(x). b) For d = 2, test the fixed rules for integration over a triangle (Section 10.4[A]) on the Dirichlet density f (t). c) Compare other integration methods on these test problems.