A biologist would like to estimate the size of a certain population of fish. A sequential approach is proposed whereby a member of the population is sampled at random, tagged and then returned. This process is repeated until a member is drawn that has been previously tagged. If desired, we could then begin tagging again with a new kind of tag. Let M be the trial at which the first previously tagged fish is sampled and N be the total population size. This process can be described in terms of a Markov chain where Xk is the number of successive untagged members observed. That is Xk = k for k = 1, 2… M – 1 XM = 0.
(a) For a fixed N = n, find the form of the transition probability matrix.
(b) Find for Pr (M = m | X0= 0) for m = 2, 3, 4… n.