New results on a generalized coupon collector problem using Markov chains

TL;DR

This paper analyzes a generalized coupon collector problem with a null coupon, deriving distribution and moments of collection time, and proves that an almost-uniform distribution minimizes expected collection time, extending to a new conjecture.

Contribution

It introduces a generalized model with a null coupon, derives key distribution properties, and proves an optimality conjecture for almost-uniform distributions.

Findings

Derived explicit distribution and moments of collection time.

Proved the almost-uniform distribution minimizes expected collection time.

Extended the result to the full collection and proposed a new conjecture.

Abstract

We study in this paper a generalized coupon collector problem, which consists in determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions of the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which minimizes the expected time to get a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of that time when the full collection is considered, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to get any fixed number of distinct coupons.