(b) Derive the moment generating function of \( X \). [A detailed derivation is required.]

(c) On the average, how many boxes of tea must be purchased by a customer to obtain a complete collection consisting of the 6 nautical figurines.

6. (10%) Let $f(x) = \frac{1}{6}, 0 \le y \le 4, y \le x \le y + 2$ be the joint pdf of $X$ and $Y$. (a) Find the conditional pdf of \( X \) given \( Y = y \) and of \( Y \) given \( X = x \).

(b) Find the conditional expectation of \( X \) given \( Y = y \).

7. (5%) Let $X_1$ and $X_2$ have the joint probability density function $$f(x_1, x_2) = \frac{1}{2}, 0 < x_1 < x_2 < 2$$ Let $Y = \frac{X_1}{X_2}$. Find the probability density of $Y$.

8. (10%) Two components operate in parallel in a device. The device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with the probability density function. The cost of operating the device is $Z = Y_1 + 2Y_2$, where $Y_1 = \min(X_1, X_2)$ and $Y_2 = \max(X_1, X_2)$ (a) Find the joint probability density function of \(Y_{1}\) and \(Y_{2}\).

(b) Find E[Z].

9. (10%) In a college health fitness program, let $X$ denote the weight in kilograms of a male freshman at the beginning of the program and $Y$ denote the weight change during a semester. Assume $X$ and $Y$ have a bivariate normal distribution with $\mu_X = 75$ and $\mu_Y = 5$, $\sigma_X^2 = 64$, $\sigma_Y^2 = 2.56$ and $\rho = -0.6$. The joint probability density function of a bivariate normal distribution is defined as $$f(x, y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\left(-\frac{q(x, y)}{2}\right)$$ where $q(x, y) = \frac{1}{1-\rho^2} \left[ \left(\frac{x-\mu_X}{\sigma_X}\right)^2 - 2\rho \left(\frac{x-\mu_X}{\sigma_X}\right) \left(\frac{y-\mu_Y}{\sigma_Y}\right) + \left(\frac{x-\mu_Y}{\sigma_Y}\right)^2 \right]$. (a) Derive the marginal pdf of \( Y \) and find \( P[2.5 \leq Y \leq 6.5] \). [A detailed derivation is required.]

(b) Derive the conditional pdf of \( Y \) given \( X = x \) and find $P[2.5 \leq Y \leq 6.5 | X = 80]$. [A detailed derivation is required.]

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