A poisson approximation in an urn model with indistinguishable balls

Abstract

Let n indistinguishable balls be distributed in m urns such that all ( m+ n−1 m−1 ) distributions are equally probable, and consider the number M k of urns containing k balls each ( k≥0). If m→∞ and n m→0 or ∞, the distribution of M k can be approximated by a suitable Poisson distribution; we obtain estimates of the errors in such an approximation to the probability function and to the distribution function of M k for large but finite m. A similar result is obtained for the model where no urn is allowed to be empty. The method used is elementary.