The generalized coupon collector problem

Neal, Peter John (2008) The generalized coupon collector problem. Journal of Applied Probability, 45 (3). pp. 621-629. ISSN 0021-9002

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Abstract

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.

Item Type:
Journal Article
Journal or Publication Title:
Journal of Applied Probability
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
76918
Deposited By:
Deposited On:
27 Nov 2015 13:52
Refereed?:
Yes
Published?:
Published
Last Modified:
18 Nov 2020 03:24