課程名稱︰統計學與實習一
課程性質︰經濟系必修
課程教師︰陳旭昇
開課學院：社科學院
開課系所︰經濟系
考試日期（年月日）︰2010/11/10
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Statistics I: Midterm Exam (2010.11.10) Department of Economics
prof. Shoi-Sheng Chen National Taiwan University
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Answers without explanation or calculation earn no point.
Problem 1 Let A and B be indipendent events with probabilities
0 < P(A) < 1 and 0 < P(B) < 1. Determine whether the followings pairs of
events are independent or not.
1.(5%)A^c and B.
2.(5%)A^c∩B and A.
3.(5%)A^c and B^c.
4.(5%)A^c∩B^c and A∪B.
Problem 2 Let {X,Y} has a bivariate normal distribution:
┌ ┐ ┌ ┐ ┌ ┐
│X│~ N(│3│ │1 -1│).
│Y│ │1│,│-1 4│
└ ┘ └ ┘ └ ┘
Let W = 5X + 2Y + 3.
1.(5%)Find the expectation of W: E(W).
2.(5%)Find the variance of W: Var(W).
Problem 3 Definition: Y is said to be mean independent of X if and only
if E(Y|X) is a constant.
1.(5%)Show that if Y is mean independent of X, then
E(Y|X) = E(Y).
2.(5%)Show that if Y is mean independent of X, then
Y and X are uncorreleted.
Notice: There are totally 3 pages with 110 points 1
Statistics I: Midterm Exam (2010.11.10) Department of Economics
prof. Shiu-Sheng Chen National Taiwan University
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Problem 4 Note, the notation "log" indicates a natural logarithm.
1.Let X = e^z, where Z ~ N(0,1) with pdf
Φ(z) = 1
--------e^(-z^2/2)
√(2π)
X is called a standard log-normal random variable.
(a)(5%)Find the pdf of X: f(x).
(b)(5%)Find the expectation of X: E(X).
(c)(5%)Find the variance of XL Var(X).
2.(5%)Let Y = e^S, where S ~ N(E(S),Var(S)). Y is called a log-normal
random variable.
Show that logE(Y) = E(logY) + 1/2 Var(logY). (1)
Equation (1) is a property that is very useful for log-linearization
in macroeconomic theory.
3.(5%)Let W denote the wealth of a representative investor. Assume
the utility function is U(W) = c - e^(-W),
where c is a constant, and W ~ N(E[W],Var[W]). Show that the expected
utility function E[U(W)] is an increasing function of expected
wealth, E[W], and a decreasing function of the variance of wealth,
Var[W]. That is, show that E[U(W)] = f(E[W],Var[W])
d E[U[W]] d E[U(W)]
and --------- > 0, and ---------- < 0.
d E[W] d Var[W]
This result justifies the use of mean-variance expected utility
functions in financial economics.
Notice: There are totally 3 pages with 110 points 2
Statistics I: Midterm Exam (2010.11.10) Department of Economics
prof.Shiu-Sheng Chen National Taiwan University
----------------------------------------------------------------------------
Problem 5 Let X ~ Bernoulli(0,3).
1.(5%)Find E(100X^100 - 200X^200)
2.(5%)Let Y = 2(X-1/2), find the pmf of Y: f(y).
Problem 6 Let {X1,X2} ~ i.i.d. Uniform(a,b).
1.(5%)Let Y = max(X1,X2). That is, Y is the maximum value among X1 and
X2. Find the CDF of Y: F(y).
2.(5%)Let W = min(X1,X2). That is, W is the minimum value among X1 and
X2. Find the CDF of W: F(w).
Problem 7 陳家村有一位鐵匠，他所生產的「金剛圈」，不但手工精美，且
價錢公道、童叟無欺。已知該鐵匠所生產的每個「金剛圈」有1-p, 0<p<1的機率是
良品，但卻有p的機率會是「尺寸太差，前重後輕，左寬右窄，使人戴上之後很不
舒服，整晚失眠」的劣質品。*1 假設每次所生產「金剛圈」的品質相互獨立。
試回答下列問題。
1.(15%)定義隨機變數Xi，若唐三藏所購買的第i個「金剛圈」為良品時，
Xi = 0，若為劣質品時Xi = 1。假定唐三藏一共買了三個「金剛圈」，隨
機變數Y = 3
Σ Xi 表示該三個「金剛圈」中為劣質品的個數。請寫出Y的
i=1
機率分配函數pmf，其分配的名稱，以及E(Y)。
2.(5%)假設該鐵匠製作一個「金剛圈」要花的時間為T，T ~ exp(θ)，其
pdf為
f(t) = 1/θ e^(-t/θ), for t≧0
0 , o/w
請問唐三藏下了三張訂單後，預期要多久才拿的到貨。
3.(5%)若唐三藏改行，批了N個「金剛圈」到天竺去賣。已知N ~ Poisson(λ)
，試求 N
E( Σ Xi) = ?
i=1
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*1語出"齊天大聖西遊記之仙履奇緣", Stephen Chou, 1995。
Notice: There are totally 3 pages with 110 points 3
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