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 ---------------------------------------------------------------------------- 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 ------------------------------------------------------------------------------ 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 --------------------------------------- *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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