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

(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 \( W \)?

(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?

(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)

(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%)