课程名称︰计量经济理论一 课程性质︰必修 课程教师︰李宗颖 开课学院：社会科学院 开课系所︰经济学研究所 考试日期（年月日）︰2014/11/12 考试时限（分钟）：90分钟 试题 : 1. Let X1 and X2 have the joint pdf f(x1,x2)=6x2 , 0<x2<x1<1 =0 , elsewhere (a)(8 points)Find the conditional probability density function (pdf) of X2 given X1=x1. (b)(4 points)Find E(X2|X1=x1). 2.(8 points) Let X have the pdf : f(x)=(1/4)*(1+x) , -2<x<2. Find the pdf of Y=X^2. 3. Let X1,X2,...,Xn be independent Bernoulli(p) random variables and n let Yn=(1/n)ΣXi i=1 (a)(10 points)Find the asymptotic distribution of Yn. (b)(10 points)Let p≠(1/2). Find the asymptotic distribution of Yn(1-Yn). 4.(20 points) The score function is defined as s(θ)=∂log(f(X|θ)/∂θ where f(X|θ)is the density function of X given parameter θ. Under the regularity conditions, E(s(θ))=0, Show that E(s(θ)^2)=-E(∂s(θ)/∂θ) 5. Let X1,X2,...,Xn be i.i.d with (marginal) pdf f(x|θ)=θx^(θ-1),0≦x≦1,0<θ<∞. (a)(10 points)Find the MLE of θ. (b)(10 points)Find the method of moments estimator of θ. 6. Suppose X1,X2,...,Xn are independent with pdf's f(xi|θ)=λi*e^(-λi*xi),θ=(λ1,λ2,...,λn)'. We want to test H0: λ1=λ2=...=λn. & H1: λi are not all equall. (a)(15 points)Find the likelihood-ratio test statisitc ξLR. (b)(5 points)What is the decision rule using ξLR with significance level α=5% (Be clear on the critical value of the test.) --

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