1. **Problem statement:** We have three scenarios involving random variables G, H, and Z with different distributions. 2. **Scenario (a):** - Random variable: $G$ = time in minutes until next geyser eruption. - Distribution: Exponential distribution because time between events in a Poisson process is exponential. - Parameter: Rate $\lambda = 4$ eruptions per hour = $\frac{4}{60} = \frac{1}{15}$ per minute. - Mean: $E(G) = \frac{1}{\lambda} = 15$ minutes. - Variance: $Var(G) = \frac{1}{\lambda^2} = 15^2 = 225$ minutes$^2$. 3. **Scenario (b):** - Random variable: $H$ = number of hits per day on fansite. - Distribution: Poisson distribution because hits occur independently with a known average rate. - Parameter: $\lambda = 2.5$ hits per day. - Mean: $E(H) = \lambda = 2.5$. - Variance: $Var(H) = \lambda = 2.5$. 4. **Scenario (c):** - Random variable: $Z$ = number of Zubat encounters in 75 steps. - Distribution: Binomial distribution because each step is independent with a fixed probability of success. - Parameters: number of trials $n=75$, probability of success $p=0.6$. - Mean: $E(Z) = np = 75 \times 0.6 = 45$. - Variance: $Var(Z) = np(1-p) = 75 \times 0.6 \times 0.4 = 18$. **Summary:** - $G \sim \text{Exponential}(\lambda=\frac{1}{15})$, $E(G)=15$, $Var(G)=225$. - $H \sim \text{Poisson}(\lambda=2.5)$, $E(H)=2.5$, $Var(H)=2.5$. - $Z \sim \text{Binomial}(n=75, p=0.6)$, $E(Z)=45$, $Var(Z)=18$.