The Coupon-Collector Problem Revisited

A standard combinatorial problem calls to cstlmate the expected number of purchases of coupons needed LO complete Ute collection of all possible m different types. Generalizing this problem. by letting lhe coupons be obtained with an arbitrary probability distribution. and considering other related processes, the problem has been found to model many practical siwations. The usefulness of lhis model has been seriously hampered by !.he computational difficulties in obtaining any numerical results concerning moments or distributions. We show, following Flajolet et al. [15], !hat Ute calculus of generating functions over regular languages may be applied La the problem, answer numerous questions about the sampling process and demonstrate their numerical efficiency. We also present a proof of a long-standing folk-theorem. concerning lhe extremalily of uniform reference probabilities. The paper concludes with a discussion of estimation problems related lo the engineering applications of lhis problem.