A standard technique used by ?eld biologists to estimMe animal populations is the capture-recapture method. To be concrete, We a lake containing an unknown number N of ?sh. A sample of size M of these ?sh is caught, tagged, and released back into the lake. A while later, another sample of size S is taken with replacement, and it is found that X of them are tagged.
(a) What is the probability distribution of X?
(b) Discuss the assumptions on the nature of ?sh and the procedure for sampling that you used to justify the distribution above. (For instance, what if tagging a ?sh is so traumatic that it dies soon thereafter?)
(c) What is the Maximum Likelihood Estimator of N ?
(d) What is the Method of Moments Estimmcr of N?
(e) Are there problems with these estimators? If so, what might you do? (f) (20 pts) What would change in your analysis if the second sample had been taken without replacement?