problem 87

(a): Let $I_{j}$ be the indicator random variable for $j$ person in the sample being a democrat. Let $X$ be the total number of democrats in the sample. Then, $E (X) = \sum_{j = 1}^{c} I_{j} = c \frac{d}{100}$
(b): Let $I_{j}$ be the indicator random variable for state $j$ being represented by at least one person in the sample. Then $P (I_{j} = 1) = 1 - \frac{(\frac{98}{c})}{(\frac{100}{c})}$ . Then, the expected number of states represented in the sample is $E (X) = 50 (1 - \frac{(\frac{98}{c})}{(\frac{100}{c})})$ .
(c): Similarly to part $b$ , $E (X) = 50 \frac{(\frac{98}{c - 2})}{(\frac{100}{c})}$ .
(d): $P (X = k) = \frac{(\frac{50}{k}) (\frac{50}{20 - k})}{(\frac{100}{20})}$ for $0 \leq k \leq 20$ .
Letting $I_{j}$ be the indicator variable for person $j$ in the sample being a junior senator of a state, $E (X) = 20 \frac{50}{100} = 10$ .
(e): Similar to part $b$ , $E (X) = 50 \frac{(\frac{98}{18})}{(\frac{100}{20})}$ .