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The number of K-tons in the coupon collector problem

Part of: Combinatorial probability Distribution theory - Probability

Published online by Cambridge University Press:  07 February 2023

John C. Saunders Open the ORCID record for John C. Saunders [Opens in a new window]
John C. Saunders*
Affiliation:
Middle Tennessee State University
*
*Postal address: Middle Tennessee State University Department of Mathematical Sciences. Email: John.Saunders@mtsu.edu
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Abstract

Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$, and that we keep buying boxes until we collect at least m coupons of each type. For $k\geq m$ call a certain coupon a k-ton if we see it k times by the time we have seen m copies of all of the coupons. Here we determine the asymptotic distribution of the number of k-tons after we have collected m copies of each coupon for any k in a restricted range, given any fixed m. We also determine the asymptotic joint probability distribution over such values of k, and the total number of coupons collected.

Keywords

Probability distributionscombinatorial probability

MSC classification

Primary: 60E05: Distributions
Secondary: 60C05: Combinatorial probability
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 723 - 736
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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