Journal of Applied Mathematics & Bioinformatics

How many trials does it take to collect all different types of a population with probability p ?

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  •                                                     Abstract

    Coupons are collected one at a time (independently and with replacement) from a population containing N distinct types. This process is repeated until all N different types (coupons) have been collected (at least once). Recently, interesting results have been published regarding the asymptotics of the moments and the variance, of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N--- 00 . Moreover, the limit distribution of the random variable TN (appropriately normalized), has been obtained for a large class of coupon probabilities (see, [9], [10], and [11]). This classical problem of probability theory has found a plethora of applications in many areas of science, and quite recently, it has been highly involved with cryptography. In this note we take advantage of the above results and present in detailed various examples that illustrate problems similar to those one faces in the real life. We also conjecture on the minimum of the variance V [TN].