The collector's problem with group drawings

Abstract

We consider sampling with replacement of equiprobable groups of a fixed size m from a finite population S. Given a subset A ⊂ S, the distributions of (a) the number of distinct elements of A in a sample of size k and (b) the sample size necessary to obtain at least say n elements of A are given. Neat formulas are given especially for the expected values of these, as well as of some related random variables. Further we derive an optimal strategy to collect all elements of S under the assumptions that sampling one group costs α monetary units and that it is possible to purchase the elements which are missing at the end of the sampling procedure at a price of β > α/m per element.