3. (10%) Let $X$ have the probability mass function

$$f_X(x) = P[X = x] = \frac{3}{2} \left(\frac{1}{2}\right)^{2|x|}, x = \pm 1, \pm 2, \dots$$

(a) \((5\%)\) Find \(P[X = 1,2,3,\dots ]\)

(b) (5%) Find \( E[X] \) and \( Var(X) \). [A detailed derivation is required and the answer has to be simplified.]

4. (10%) Suppose that 500 points are selected independently and at random from the unit square $$\{(x, y): 0 \le x < 1, 0 \le y < 1\}.$$ Let $W$ equal the number of points that fall into $\{(x, y): x^2 + y^2 < 1\}$. (a) (5%) Write down the distribution of \( W \) and the corresponding probability mass function.

(b) (5%) Find the mean and variance of \( W \). [A detailed derivation is required.]

5. (15%) Red Rose tea randomly began placing 1 of 6 English porcelain miniature figurines in a 100-bag box of the tea, selecting from 6 nautical figurines, which are numbered 1 to 6. (a) Let \( X \) denote the number of Red Rose tea that a family would have to buy to have at least three of the nautical figurine having number 1. Write down the probability mass function of \( X \).

(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 \(Y_{2}\).

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