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

(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 \( U \).

(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.

(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 \( \operatorname{Cov}(X) \).

(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 \( \bar{X}_n \).