problem 60

(a): By the story of the problem, $X \sim NHGeom (n, N - n, m)$ . Then, $Y = X + m$ .
(b): According to part $a$ , $E (Y) = E (X) + m = \frac{m (N - n)}{n + 1} + m$ . The implied indicator variables are the same as in the proof of the expectation of Negative Hypergeometric random variables.
(c): The problem can be modeled with a Hypergeometric random variable $Z \sim HGeom (n, N - n, E (Y))$ . Then, $E (Z) = E (Y) \frac{n}{N} = (\frac{m (N - n)}{n + 1} + m) \frac{n}{N} = m \times \frac{N + 1}{n + 1} \times \frac{n}{N}$ . Since $\frac{N + 1}{n + 1} \times \frac{n}{N} < 1 \Rightarrow (n + 1) N (n - N) < 0 \Rightarrow \frac{N - n}{(n + 1) N} > 0 \Rightarrow n < N$ for positive $n$ and $N$ , $E (Z) < m$ .