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New Results on a Generalized Coupon Collector Problem Using Markov Chains

Part of: Combinatorial probability Markov processes

Published online by Cambridge University Press:  30 January 2018

Emmanuelle Anceaume ,
Yann Busnel  and
Bruno Sericola
Emmanuelle Anceaume*
Affiliation:
CNRS
Yann Busnel*
Affiliation:
Université de Nantes
Bruno Sericola*
Affiliation:
INRIA
*
∗ Postal address: CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France.
∗∗ Postal address: Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France.
∗∗∗ Postal address: INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France. Email address: bruno.sericola@inria.fr
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Abstract

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In this paper we study a generalized coupon collector problem, which consists of 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 for the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which minimizes the expected time to obtain a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of the time needed to obtain the full collection, 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 obtain any fixed number of distinct coupons.

Keywords

Coupon collector problemMinimizationMarkov chain

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60J05: Discrete-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 405 - 418
Copyright
© Applied Probability Trust 

References

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