Derivation of a joint occupancy distribution via a bivariate inclusion and exclusion formula

Abstract

Consider a supply of balls randomly distributed into n distinguishable urns and assume that the number of balls distributed into any specific urn is a random variable with probability function . The joint probability function and binomial moments of the number K _i of urns occupied by i balls each and the number K _j of urns occupied by j balls each, i≠j, given that a total of S _n =m balls are distributed into the n urns, are derived in terms of convolutions of q _x , x=0,1, . . . and their finite differences. Also, some illustrating examples are discussed.