課程名稱︰高等統計推論
課程性質︰選修∕數研所統計組必修
課程教師︰陳宏 教授
開課系所︰數學系
考試時間︰
試題：
Instructions. You have 60 minutes to complete the examination. This is a
closed-book examination.
1. (20%) Let Y＝aX＋b＋ε where a and b are constants. Here X is a normal
random variable with mean μ and variance σ^2 and ε is also a normal
random variable with mean 0 and variance σ^2. When X and ε are indepen-
dent, find Cov(X,Y).
2. It is known that both X and Y are normally distributed with mean 0 and
variance 1. Set
╭ X, if XY＞0,
Z＝│ 0, if XY＝0,
╰ -X, if XY＜0.
(a) (20%) Show that Z is normally distributed.
(b) (10%) someone claims that (Z,Y) has the bivariate normal pdf. Do you
agree to that claim? Give reason to justify your answer.
Hint: Check on the sign of ZY.
3. (35%) It is known that
X_i|P_i～Bernoulli(P_i), i＝1,...,n,
P_i～Beta(α,β).
n
Consider Y＝ Σ X_i.
i=1
(a) (15%) Determine E(Y) and Var(Y). 2
Fact: E(P_i)＝α∕(α＋β) and Var(P_i)＝αβ∕[(α＋β) (α＋β＋1)]
(b) (10%) What is the distribution of Y?
(c) (10%) What is the distribution of P_i|X_i?
4. (15%) Determine the distribution of Z＝X_1X_2 where X_1, X_2 are indepen-
dent UNIF(0,1) random variables.
Hint: You may want to find the distribution of logX_1 first.
Some solutions or hints:
2
1. aσ .
2. (a) Use conditional probabilty.
(b) No.
3. (a) E(Y)＝ΣE(X_i) and E(X_i)＝E(P_i),
Var(Y)＝ΣVar(X_i) and Var(X_i)＝E(P_i)－(E(P_i))^2.
(b) Bin(n,α∕(α＋β)).
(c) Beta(X_i＋α,β－X_i＋1).
4. Use logZ＝logX_1＋logX_2 and logX_i～Exp(1).
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