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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）︰2007/01/16
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 1: Final Exam (January 16, 2007)
Note: Answers without explanation or calculation earn no point
[試題開始]
Problem 1 (2, 2, 1, 1 pts) Let Rk and Rm denote the percentage returns on
asset K and market portfolio, repectively. Let the joint pmf. of Rk and Rm
be defined by
f(rk,rm) = (rk + rm) / 21 ,
with supports supp(Rk) = {1,2} and supp(Rm) = {1,2,3}
1. Find the marginal pmf of Rk and Rm: f(rk), f(rm).
2. Find the MGF of Rk and Rm: Mk(t), Mm(t).
3. Find out the β of asset K.
4. Find out the risk-free rate rf.
Problem 2 (2, 2, 2 pts) Let X~N(0,1) and Y = e^X
1. What is the support of Y?
2. Find out the pdf of Y.
3. Find out E(Y) and Var(Y).
Problem 3 (2, 2, 2 pts) Let Xn~Binomial(n,μ) and Yn = Xn/n
p
1. Suppose that Yn─→q. Find out the value of q.
2. Find out the asymptotic distribution of Yn.
3. Use Delta's Method to find out the asymptotic distribution of (Yn)^2
Problem 4 (2, 2, 2 pts) Let X,Y,Z be mutually independent random variables
with Poisson distribution having means 4, 3, 2, repectively.
1. Find the moment-generating funtion of the sum W = X+Y+Z.
2. How is W distributed?
3. Suppose the distribution function of W is denoted by F(w). Now assume that
{Wi}100 a random sample from F(w). Use CLT to approximate the probability
i=1
100
__
P( 900 ＜ ＞ (Wi) ＜ 960 ) p.s.中間那個醜醜的是summation(sigma)
￣ ￣ ￣
i=1
Problem 5 (2, 2, 2 pts) Let {Xi} n be a random sample from N(5,15). Define
i=1
the sample mean and sample variance as
__ __
Xn = (＞i Xi) / n ,
￣
__ _ __ _
S^2 = ＞i(Xi-Xn)^2 / (n-1) = ＞i(Xi)^2 - n*(Xn)^2 / (n-1)
￣ ￣
We also define
~ __ _ __ _
S^2 = ＞i(Xi-Xn)^2 / n = ＞i(Xi)^2 - n*(Xn)^2 / n
￣ ￣
~
1. Find out E(S^2) and E(S^2).
~
2. Find out Var(S^2) and E(S^2).
~ p
3. Suppose that S^2 ─→ k. Find out the value of k.
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