The probability generating function of empty cell variable in a randomized occupancy problem

Abstract

Let MO denote the number of empty cells when n distinguishable balls are distributed independently and at random in m cells such that each ball stays with probability p in its cell, and falls through with probability 1–p. We find the probability generating function of MO by solving a partial differential equation satisfied by a suitable generating function. The corresponding function for the classical case p = 1 is well-known, but obtained by different methods. © 1985, Taylor & Francis Group, LLC. All rights reserved.