Suppose that a family of four wants to estimate the number of goldfish in its pond. Call this number N. The family decides to use a capture/recapture strategy for collecting data. They plan to capture a random sample of A fish, mark them so they can be identified later, and then release them back into the pond. After allowing sufficient time for these fish to circulate with the other fish, they will then capture a new, independent random sample of n goldfish and observe how many of those n goldfish are marked. Let the random variable X represent the number of goldfish captured in the second sample that are marked as having also been caught in the first sample.
(a) Derive an expression for the method-of-moments estimator of N, in terms of A, n, and X. Now suppose that the family captures 10 goldfish in the first sample and 12 goldfish in the second sample, of which 5 had been marked.
(b) Determine the method-of-moments estimate of N.