Bivariate distributions generated from Pólya-Eggenberger urn models

In several classic papers, Pólya and Eggenberger used urn models to generate distributions that could be used to model contagion processes. Three bivariate versions are investigated together with their limiting distributions. In this way a large class of bivariate distributions (beta, Pareto, gamma, negative binomial, Poisson, binomial) is obtained. These derivations also show connections between the various distributions. In particular, some limiting distributions are shown to be upper Fréchet bounds, and some distributions are associated.