44. A lake has N trout, and t of them are caught at random, tagged, and returned. We catch n trout at a later time randomly and observe that m of them are tagged.

(a) Find PN , the probability of what we observed actually happen.

(b) To estimate the number of trout in the lake, statisticians find the value of N that maximizes PN . Such a value is called the maximum likelihood estimator of N. Show that the maximum of PN is[nt/m], where by [nt/m] we mean the greatest integer less than or equal to nt/m. That is, prove that the maximum likelihood estimator of the number of trout in the lake is [nt/m].