Sums, products, and ratios of non-central beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products, and ratios of independent beta random variables. In this article, we extend Professor Pham-Gia's results when X-1 and X-2 are independent random variables distributed according to the non-central beta distribution of Type I, the non-central beta distribution of Type H, and the doubly non-central beta distribution. For each of these three distributions, we derive exact expressions for the densities of S = X-1 + X-2, D = X-1 - X-2, P = X1X2, and R = X-2/X-1. The expressions turn out to involve the hypergeometric functions of one and two variables.