An examination of the negative occupancy distribution and the coupon-collector distribution

TL;DR

This paper analyzes the negative occupancy and coupon-collector distributions, providing insights into their properties, computation methods, and practical approximation techniques within the context of generalized occupancy problems.

Contribution

It offers a comprehensive examination of these distributions, including their properties, computational approaches, and guidance on approximation for large-scale applications.

Findings

Distribution properties are characterized and analyzed.

Methods for computing and approximating the distributions are provided.

Practical guidelines for large-scale computation and approximation are discussed.

Abstract

We examine the negative occupancy distribution and the coupon-collector distribution, both of which arise as distributions relating to hitting times in the extended occupancy problem. These distributions constitute a full solution to a generalised version of the coupon collector problem, by describing the behaviour of the number of items we need to collect to obtain a full collection or a partial collection of any size. We examine the properties of these distributions and show how they can be computed and approximated. We give some practical guidance on the feasibility of computing large blocks of values from the distributions, and when approximation is required.