3.2.15 problem 34

(a)

Let $Y$ be the number of Statistics students in the sample of size $m$.

$$P(Y=k)=\sum _{i=k}^{n}P(X=i)P(Y=k|X=i)=\sum _{i=k}^n\left(\frac{n}{i}\right)p^i{(1-p)}^{n-i}\frac{\left(\frac{i}{k}\right)\left(\frac{n-i}{m-k}\right)}{\left(\frac{n}{m}\right)}$$

(b)

Consider a student in a random sample of size $m$. Independently of other students, the student has probability $p$ of being a statistics major. Then, the probability of $k$ students in the sample being statistics majors is $\left(\frac{m}{k}\right)p^k{(1-p)}^{m-k}$. Thus, $Y\sim \mathrm{Binom}(m,p)$.