We study in this paper a generalized coupon collector problem, which consists
in determining the distribution and the moments of 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 obtain expressions of the distribution and the moments of this time. We also
prove that the almost-uniform distribution, for which all the non-null coupons
have the same drawing probability, is the distribution which minimizes the
expected time to get a fixed subset of distinct coupons. This optimization
result is extended to the complementary distribution of that time when the full
collection is considered, proving by the way this well-known conjecture.
Finally, we propose a new conjecture which expresses the fact that the
almost-uniform distribution should minimize the complementary distribution of
the time needed to get any fixed number of distinct coupons.Comment: 14 page
