苏良军高等数理统计习题答案.pdf

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1、AdvancedMathematicalStatisticsChapter11Provethefollowing:(a)IftheeventsAandBareindependent,thensoareAcandBc.(b)TheeventsΩand∅areindependentofanyeventA.(c)Isthereanyeventwhichisindependentofitself?Ifso,giveanexample.Answer.(a)ThestatedresultisimmediatefromP(Ac∩Bc)=P((A∪B)c)=1−P

2、(A∪B)=1−P(A)−P(B)+P(A∩B)=1−P(A)−P(B)+P(A)P(B)=(1−P(A))(1−P(B))=P(Ac)P(Bc).(b)ForanyeventA,wehaveP(A∩Ω)=P(A)=P(A)P(Ω),andP(A∩∅)=P(∅)=P(A)P(∅)=0,asdesired.(c)Yes.Fromthedefinitionofindependence,aneventAisindependentofitselfifP(A∩A)=P(A)P(A),i.e.,P(A)=1,or0.Thus,Ωand∅areobviousexa

3、mplesofsuchanevent.3LetAandBbetwoeventswithprobabilitiesP(A)=1/2,P(B)=2/3,andP(A∩B)=1/3.DefinerandomvariablesX=1(A)andY=1(B),where1(A)=1ifAistrueand0otherwise.(a)FindthejointdistributionofXandY.(b)ObtaintheconditionaldistributionofXgivenY.Answer.(a)ThejointdistributionofXandYis

4、givenby1P(X=1,Y=1)=P(A∩B)=,3c111P(X=1,Y=0)=P(A∩B)=P(A)−P(A∩B)=−=,236c211P(X=0,Y=1)=P(A∩B)=P(B)−P(A∩B)=−=,333P(X=0,Y=0)=P(Ac∩Bc)=1−P(A∪B)=1−P(A)−P(B)+P(A∩B)1211=1−−+=.2336(b)TheconditionaldistributionofXgivenYfollowsas1P(X=1,Y=1)P(A∩B)1P(X=1

5、Y=1)===3=,P(Y=1)P(B)22311P(X=0

6、Y=1)=

7、1−P(X=1

8、Y=1)=1−=,22P(X=1,Y=0)P(A∩Bc)11P(X=1

9、Y=0)===6=,P(Y=0)P(Bc)12311P(X=0

10、Y=0)=1−P(X=1

11、Y=0)=1−=.22R∞4.LetY≥0bearandomvariableandp>0.ProveE(Yp)=pyp−1P(Y>y)dy.0Answer.1LetY˜FY.ThenZ∞Z∞Z∞pyp−1P(Y>y)dy=pyp−1dF(Y)dyY00yZ∞ZY=pyp−1dydFY00Z∞=YpdF=E(Yp).Y06Thejointprobabilitydensityf

12、unctionofXandYisgivenbyf(x,y)=3(x+y)1{0≤x+y≤1,0≤x,y≤1}.(a)FindthemarginaldensityofX.(b)FindP{ω

13、X(ω)+Y(ω)<1/2}.(c)FindE(Y

14、X).(d)FindCov(X,Y).Answer.(a)ThemarginaldensityofXisgivenby(R¡¢1−x323(x+y)dy=1−x,if0≤x≤1f(x)=02.0otherwise(b)Z1Z1−x221P{X+Y<1/2}=3(x+y)dydx=.008(c)Thecondit

15、ionaldensityofYgivenX=xis3(x+y)2(x+y)f(y

16、x)==,3(1−x2)(1−x2)2for0≤x+y≤1,0≤x,y≤1.SoZ1−x2(x+y)(1−x)(x+2)E(Y

17、X=x)=ydy=,0(1−x2)3(1+x)(1−X)(X+2)implyingthatE(Y

18、X)=.3(1+X)R1¡¢(d)E(X)=3x1−x2dx=3.Byasymmetricargument,E(Y)=3.Next,0288Z1Z1−x1E(XY)=3xy(x+y)dydx=.0010Consequently,13313Cov(

19、X,Y)=E(XY)−E(X)E(Y)=−=−.10883201.7LetXandYhavethejointdensity