Solution
Fifty Tagged Fish and a Tie
Show the problem again
A lake contains an unknown number N of fish. Fifty are caught, tagged and released. After mixing, a second sample of 40 fish is drawn at random without replacement, and 8 of them carry tags. Treat the number of tagged fish in the second sample as hypergeometric. The likelihood of the data, as a function of the integer N, attains its maximum at exactly two values of N. What is the smaller of the two?
Worked solution
The likelihood is L(N) = C(N−50, 32)/C(N, 40) up to a constant. Compare consecutive values: L(N)/L(N−1) = (N−50)(N−40) / (N(N−82)), where the 82 is 50 + 40 − 8. This ratio is at least 1 exactly when (N−50)(N−40) ≥ N(N−82), which simplifies to 2000 ≥ 8N, i.e. N ≤ 250. Equality holds at N = 250, so L(250) = L(249) and these are the two maximizers. Here 50·40/8 = 250 is an integer; the familiar Lincoln-Petersen estimate Kn/k lands exactly on the tie, and the smaller maximizer is 249.
Source: Lincoln-Petersen capture-recapture setting; the exact-likelihood tie is a standard remark in the literature. Statement written for AxiomIQ.