Question: Elk dwell in a certain forest. There are N elk, of which a simple random sample of size n is captured and tagged (so all N n sets of n elk are equally likely). The captured elk are returned to the population, and then a new sample is drawn. This is an important method that is widely used in ecology, known as capture-recapture. If the new sample is also a simple random sample, with some fixed size, then the number of tagged elk in the new sample is Hypergeometric. For this problem, assume that instead of having a fixed sample size, elk are sampled one by one without replacement until m tagged elk have been recaptured, where m is specified in advance (of course, assume that 1 ≤ m ≤ n ≤ N). An advantage of this sampling method is that it can be used to avoid ending up with a very small number of tagged elk (maybe even zero), which would be problematic in many applications of capture-recapture. A disadvantage is not knowing how large the sample will be.
(a) Find the PMFs of the number of untagged elk in the new sample (call this X) and of the total number of elk in the new sample (call this Y).
(b) Find the expected sample size EY using symmetry, linearity, and indicator r.v.s