清华经管计量经济学习题1.pdf

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1、ProblemSet1Dueon03/17/2014,BEFORE9:50am1YouarerequiredtowriteyoursolutionsinEnglish.Pleaseshowyourwork.Youdonotneedtocopythequestions.1.FindE(E(E(Y

2、X1,X2,X3)

3、X1,X2)

4、X1).
2.IfE(Y

5、X)=a+bX,findE(YX)asafunctionofE(X)andEX2.3.SupposethattherandomvariablesYandXonlytakethevalue

6、s0and1,andhavethefollow-ingjointprobability(mass)distributionX=0X=1Y=00.10.2Y=10.40.3(a)FindthemarginalpmfforX.(b)FindthemarginalpmfforY.(c)FindthemarginalCDFforX.

(d)FindE(Y

7、X=0),EY2

8、X=0,E(Y

9、X=1)andEY2

10、X=1.4.ProofthatVar(X+b)=Var(X)foracontinuousrandomvariableXwithpd

11、ff(·).5.SupposethattherandomvariableXhavethefollowingpmfValuesofX-101Probability0.30.40.3andY=X2.(a)FindE(X)andE(Y).(b)ShowthatCov(X,Y)=0.(c)ShowthatXandYarenotindependent.(d)Whatdoyoulearnfromthisexercise?6.Supposethatyouareaskedtoconductastudytodeterminewhethersmallerc

12、lasssizesleadtoimprovedstudentperformanceoffirst-yearhighschoolstudents.(a)Ifyoucouldconductanyexperimentyouwant,whatwouldyoudo?Bespecific.(b)Morerealistically,supposeyoucancollectobservationaldataonseveralthousandfirst-yearhighschoolstudentsinagivencity.Youcanobtainthesize

13、oftheirfirst-yearhighschoolclassandastandardizedtestscoretakenattheendofthefirst-yearhighschool.Whymightyouexpectanegativecorrelationbetweenclasssizeandtestscore?1Pleasebringyourproblemsettoclass,andsubmitittotheinstructorbeforethelecturebegins.1(c)Wouldanegativecorrelatio

14、nnecessarilyshowthatsmallerclasssizescausebetterperfor-mance?Explain.n7.Supposeyouhaveasequenceofi.i.d.randomvariables{Xi}i=1.AndsupposethatE(X1)=µ.Xn
1DefineX¯=Xi.ProvethatEX¯=µ.ni=1n8.Supposeyouhaveasequenceofi.i.d.randomvariables{Xi}i=1.AndsupposethatVar(X1)=1Xn1Xn


15、σ2.DefineX¯=Xandσc2=X−X¯2.ProvethatEσc2=σ2.iinn−1i=1i=12