Assume that a lake contains a total of N fish. Suppose we catch r among these fish, mark them in some way, and then put them back in the lake. After a certain amount of time, so that the marked fish have mixed well with the other ones in the lake, we select randomly n fish from the lake. Let X be the number of marked fish in that sample.

(i) Find the probability function of X.

(ii) What is the expected number of marked fish that will be caught in the sample of n fish?

(The answer to Part (i) has been found already in Section 2.10, without any reference to the hypergeometric distribution. See that section for applications of this particular problem.)