Generalized Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac and Acharya-Swamy Statistics and the Polya Urn Model

Generalized probability distributions for Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics, with unequal source probabilities $q_i$ for each level $i$, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the P\'olya urn model, involving neither independent nor identically distributed ("ninid") sampling. The most probable P\'olya distribution contains the Acharya-Swamy intermediate statistic.