Composition for multivariate random variables

We show how to find mixing probabilities, or weights, for composite probability mass functions (pmfs) for k-variate discrete random variables with specified marginal pmfs and a specified, feasible population correlation structure. We characterize a joint pmf that is a composition, or mixture, of 2^k-1 extreme correlation joint pmfs and the joint pmf under independence. Our composition method is also valid for multivariate continuous random variables. We consider the cases where all of the marginal distributions are discrete uniform, negative exponential, or continuous uniform.