A Unified Derivation of Occupancy and Sequential Occupancy Distributions

Consider a supply of balls randomly distributed in n + r distinguishable urns and assume that the number X of balls distributed in any specific urn is a random variable with probability function Pr[X = j] = q _j , j = 0,1,2…. The probability function and factorial moments of the number K _i of urns occupied by i balls each, among n specified urns, given that a total of S_n+r = m balls are distributed in the n + r urns, are expressed in terms of finite differences of the u-fold convolution of q _j , j = 0,1,2,…. As a particular case, the probability function and factorial moments of the number K = n − K₀ of occupied urns (by at least one ball), among n specified urns, given that S_n+r = m, are deduced. Further, when balls are sequentially distributed, the probability function and ascending factorial moments of the number W _k of balls required until a predetermined number k of urns, among n specified urns, are occupied, are also expressed in terms of finite differences of the u-fold convolution of q _j , jj = 0,1, 2,…. Finally, the conditional probability function Pr[W_k+1 − W _k = j|W _k = m], j = 1,2,…, is derived. Illustrating these results, the cases with q _j , j = 0,1,2,…, the Poisson, geometric, binomial and negative binomial distributions are presented.