### A coupon collector's problem with bonuses

*Toshio Nakata, Izumi Kubo*

#### Abstract

In this article, we study a variant of the coupon collector's problem introducing a notion of a

*bonus*. Suppose that there are c different types of coupons made up of bonus coupons and ordinary coupons, and that a collector gets every coupon with probability 1/c each day. Moreover suppose that every time he gets a bonus coupon he immediately obtains one more coupon. Under this setting, we consider the number of days he needs to collect in order to have at least one of each type. We then give not only the expectation but also the exact distribution represented by a gamma distribution. Moreover we investigate their limits as the Gumbel (double exponential) distribution and the Gauss (normal) distribution.Full Text: PDF