Affiliation:
1. Faculty of Mathematics, University of Belgrade, 11000 Belgrade, Serbia
Abstract
We consider the following generalization of the classical coupon collector problem. We assume that, in addition to the initial collection of standard coupons, there is one more coupon that acts as a reset button, removing all coupons from the part of the collection that has already been drawn. For the case where standard coupons have unequal probabilities of being drawn, we obtain the distribution of the waiting time until the end of the collection process. For the case where standard coupons have equal probabilities, we derive a simple formula for the expected waiting time in terms of the beta function, and discuss the asymptotic properties of this expected waiting time, when the number of standard coupons tends toward infinity.
Funder
Ministry of Science, Technological Development and Innovation of the Republic of Serbia
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference19 articles.
1. Feller, W. (1968). An Introduction to Probability Theory and Its Applications, John Wiley and Sons Inc.. [3rd ed.].
2. Asymptotic distributions for occupancy and waiting time problems with positive probability of falling through the cells;Ann. Probab.,1974
3. Mahmoud, H. (2009). Polya urn Models, Chapman & Hall/CRC.
4. Birthday paradox, coupon collectors, caching algorithms and self-organizing search;Flajolet;Discret. Appl. Math.,1992
5. New results on a generalized coupon collector problem using Markov chains;Anceaume;J. Appl. Probab.,2015