Products of Independent Beta Variables with Application to Connor and Mosimann's Generalized Dirichlet Distribution
Ian James · Journal of the American Statistical Association · 1972
Necessary and sufficient conditions are found for the product of independent beta variables to be a beta variable, and it is shown that if Z is a beta variable of the second kind and X an independent, non-degenerate random variable taking values in (0, 1), then the product XZ is a beta variable of the second kind if and only if X is beta with appropriate parameters. These results are then applied to show that if certain ratios of proportions are assumed to have marginal beta distributions, the generality of Connor and Mosimann's generalized Dirichlet distribution is severely restricted.