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# Q 55 , Chapter 4, Introduction to probability, Joseph K. Blitzstein (page 188). Please solve this question: Elk dwell in a certain forest.

Q 55 , Chapter 4, Introduction to probability, Joseph K. Blitzstein (page 188). Please solve this 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  C(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. Hint: We can assume that even after getting m tagged elk, they continue to be captured until all N of them have been obtained; brieﬂy explain why this can be assumed. Express X = X1 +···+ Xm, where X1 is the number of untagged elk before the ﬁrst tagged elk, X2 is the number between the ﬁrst and second tagged elk, etc. Then ﬁnd EXj by creating the relevant indicator r.v. for each untagged elk in the population.

(c) Suppose that m,n,N are such that EY is an integer. If the sampling is done with a ﬁxed sample size equal to  EY rather than sampling until exactly m tagged elk are obtained, ﬁnd the expected number of tagged elk in the sample. Is it less than m, equal to m, or greater than m (for n<N)?