The Siblings of the Coupon Collector

TL;DR

This paper studies a variant of the coupon collector problem involving multiple siblings sharing coupons, developing asymptotic techniques to analyze the expected number of missing coupons for each sibling as the number of coupon types grows large.

Contribution

It introduces methods to compute asymptotics of the expected missing coupons for each sibling in a multi-generational coupon collection model with unequal probabilities.

Findings

Derived asymptotic formulas for expected missing coupons as N→∞

Analyzed various probability distributions for coupon types

Provided illustrative examples demonstrating the techniques

Abstract

The following variant of the collector's problem has attracted considerable attention relatively recently (see, e.g., N. Pintacuda 1980, D. Foata H. Guo-Niu and B. Lass 2001, D. Foata and D. Zeilberger 2003, I. Adler, S. Oren and S. Ross 2003, and S. Ross 2010): There is one main collector who collects coupons. Assume there are $N$ different types of coupons with, in general, unequal occurring probabilities. When the main collector gets a "double", she gives it to her older brother; when this brother gets a "double", he gives it to the next brother, and so on. Hence, when the main collector completes her collection, the album of the $j$-th sibling, $j = 2, 3, \dots$, will still have $U_j^N$ empty spaces. In this article we develop techniques of computing asymptotics of the average $E[U_j^N]$ of $U_j^N$ as $N \rightarrow \infty,$ for a large class of families of coupon probabilities. We also give various illustrative examples.