107年 - 107 國立臺北大學_碩士班一般入學考試試題_通訊工程學系：機率#147251

科目：研究所、轉學考(插大)◆機率 | 年份：107年 | 選擇題數：0 | 申論題數：16

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所屬科目：研究所、轉學考(插大)◆機率

選擇題 (0)

申論題 (16)

1. (5% each, totally 20%) Let X and Y be random variables, with the joint PMF:

$$P_{X,Y}(x,y) = \begin{cases} c & \text{if } (x,y) = (1,1), (1,2), (2,1), (2,2); \\ 0 & \text{else}, \end{cases}$$

where c is a constant.

(a) What is the constant \( c \) to guarantee that the joint PMF is valid?
(b) What is the PMF \(P_{X}(x)\) of \(X\)?
(c) Let \( W = \min(X, Y) \). What is the PMF of \( W \)?
(d) What is the CDF of \( W \)?
2. (5% each, totally 15%) Let X be the amount of time walking from a park to another. There are two paths between the two parks, denoted by H₁ and H₂. If path H₁ is chosen, then X is the exponential random variable, with the mean of 10 minutes. If path H₂ is chosen, then X is the exponential random variable, with the mean of 20 minutes.

(a) What is the conditional PDF \( f_{X|H_1}(x) \) given path \( H_{1} \) is chosen?
(b) What is the conditional expectation of \( X \) given path \( H_{1} \) is chosen?
(c) Suppose that a student chooses the two paths equal-likely. What is the expected value of \( X \) for the student?
3. (5% each, totally 15%) Let X and Y are two independent random variables, with the PMFs as follow.

$$P_X(x) = \begin{cases} 1/2 & \text{if } x=1,2; \\ 0 & \text{else}. \end{cases}$$

$$P_Y(y) = \begin{cases} 1/2 & \text{if } x=3,4; \\ 0 & \text{else}. \end{cases}$$

(a) What is the moment generating function of \( X \)?
(b) Let \( Z = X + Y \). What is the moment generating function of \( Z \)?
(c) What is the PMF of \( Z \)?
4. (24%) Let X be a random variable uniformly distributed over [50,100] and Y = 200/X. Let Fₓ(x) and Fᵧ(y) denote the CDFs of X and Y, respectively, and fₓ(x) and fᵧ(y) denote the PDFs of X and Y, respectively. Find Fₓ(x), Fᵧ(y), fₓ(x) and fᵧ(y). (6% each)
5. (26%) Let X and Y be jointly distributed random variables with joint PDF

$$f_{X,Y}(x,y) = \begin{cases} k, & 0 \le x \le y \le 2; \\ 0, & \text{otherwise} \end{cases}$$

where k is a constant.

(a) Find the value of \( k \). (2%)
(b) Find the marginal PDFs \( f_{X}(x) \). (6%)
(c) Find the marginal expectation E[X]. (6%)
(d) Find the conditional PDF \( f_{X|Y}(x|y) \). (6%)
(e) Find the conditional expectation E[X|Y]. (6%)