108年 - 108 國立臺北大學_日間學士班轉學生招生考試試題_統計學系日間學士班3年級：機率論#147618

科目：研究所、轉學考(插大)◆機率 | 年份：108年 | 選擇題數：0 | 申論題數：13

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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 (13)

1. (35 points) The random variable X has the moment generating function

$$M(t) = \sum_{x=0}^{4} \left( \frac{3 - |x - 2|}{k} \right) e^{tx}.$$

(a) Write down the space of X.
(b) Find k.
(c) Sketch the C.D.F. of X.
(d) Calculate EX and VarX.
(e) Calculate Pr(X < 3).
(f) Let Y = X². Find the p.m.f. of Y.
2. (30 points) Let X and Y have the joint p.d.f.

$$f(x, y) = kx^2 e^{x-y}, 0 < x < \infty, 2x < y < 3x.$$

(a) Find k.
(b) Find the marginal p.d.f. of X.
(c) Find EX.
(d) Are X and Y independent? Explain your answer.
3. (35 points) Let X and Y have the joint p.d.f.

$$f(x, y) = \frac{1}{64} xy^2 \exp\left(\frac{-(x + y)}{2}\right), \quad 0 < x, y < \infty.$$

(a) Are X and Y independent? Explain your answer.
(b) Calculate Pr(X > 3).
(c) Let Z = X/Y. Find the marginal p.d.f. of Z.