112年 - 112 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：機率概論#147570

科目：【阿摩】未分類題庫 | 年份：112年 | 選擇題數：0 | 申論題數：12

試卷資訊

所屬科目：【阿摩】未分類題庫

選擇題 (0)

申論題 (12)

1. (25%) The random variable $X$ has the moment generating function

$$M(t) = \sum_{x=1}^{S} \frac{x}{c} e^{tx}$$

(a) Find the unknown constant \( c \).
(b) Calculate \(\operatorname{E}(X)\) and \(\operatorname{Var}(X)\).
(c) Sketch the C.D.F. of \(X\), \(F_{X}(x)\).
(d) Let \( U = F_{X}(X) \). Find the C.D.F. of \( U \).
2. (25 %) An urn contents 10 balls, numbered from 1 to 10. For each game, the player will select 2 balls and will win the game, if the sum of the numbers on the balls is 10. Let $X$ be the number of games that the player needs to play for 1st win.

(a) Find the p.m.f. of \( X \) and \( \operatorname{E}(X) \) if the 2 balls are selected with replacement.
(b) Find the p.m.f. of \( X \) and \( \operatorname{E}(X) \) if the 2 balls are selected without replacement.
3. (30 %) Let random variables $X$ and $Y$ have the joint p.d.f.

$$f(x, y) = cxy^3, \quad 0 < y < x < 1$$

(a) Find the unknown constant \( c \).
(b) Find the marginal p.d.f.s of \(X\) and \(Y\).
(c) Calculate \(\operatorname{E}(X), \operatorname{E}(Y), \operatorname{Var}(X), \operatorname{Var}(Y)\) and \(\operatorname{Cov}(X)\).
4. (20 %) Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with p.d.f.

$$f(x) = cx(1 - x), \quad 0 < x < 1$$

(a) Find the unknown constant \( c \).
(b) Find the moment generating function of \( X_{i}, i = 1,2,\ldots,n \).
(c) Let \(\bar{X}_n = (X_1 + X_2 + \dots + X_n) / n\). Find the moment generating function of \(\bar{X}_n\).