  We study in this paper a generalized coupon collector problem, which consists
in analyzing the time needed to collect a given number of distinct coupons that
are drawn from a set of coupons with an arbitrary probability distribution. We
suppose that a special coupon called the null coupon can be drawn but never
belongs to any collection. In this context, we prove that the almost uniform
distribution, for which all the non-null coupons have the same drawing
probability, is the distribution which stochastically minimizes the time needed
to collect a fixed number of distinct coupons. Moreover, we show that in a
given closed subset of probability distributions, the distribution with all its
entries, but one, equal to the smallest possible value is the one, which
stochastically maximizes the time needed to collect a fixed number of distinct
coupons. An computer science application shows the utility of these results.
