On Birthday, Collectors', Occupancy and Other Classical Urn Problems

Abstract

Summary An urn contains r different balls. Balls are drawn with replacement until any k balls have been obtained at least m times each. How many draws are necessary? How many balls have been drawn exactly v times? Special cases of such problems are often named as birthday, collectors', dixie cup or occupancy problems. This paper presents a unified approach to such problems by imbedding in Poisson processes. In this way we see that many classical urn problems are closely related to properties of order statistics and extreme values from the gamma distribution. Both exact and asymptotic results are derived. Finally, a brief discussion is given on other drawing schemes.