Usefulness of Asymptotic Distributions in the Classical Occupancy Problem

Abstract

Approximations to the distribution of a discrete random variable originating from the classical occupancy problem are explored. The random variable X of interest is defined to be how many of N elements selected by or assigned to K individuals when each of the N elements is equally likely to be chosen by or assigned to any of the K individuals. Assuming that N represents the number of cells and each of the K individuals is placed in exactly one of the cells, X can also be defined as the number of cells occupied by the Kindividuals. In the literature, various asymptotic results for the distributions of X and (N − X) are given; however, no guidelines are specified with respect to their utilization. In this article, these approximations are explored for various values of K and N, and rules of thumb are given for their appropriate use.