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1、PARTONESolutionstoExercisesChapter2ReviewofProbability¢SolutionstoExercises1.(a)ProbabilitydistributionfunctionforYOutcomeY=0Y=1Y=2(numberofheads)probability0.250.500.25(b)CumulativeprobabilitydistributionfunctionforYOutcomeY<00ÇY<11ÇY<2Yí2(numberofheads)Probability00.250.751
2、.0(c)m=()EY=ì(00.25)(10.50)(20.25)1.00+ì+ì=Y22UsingKeyConcept2.3:var()YE=-(YE)[()],Yand2222EY()(=ì+ì+ì=00.25)(10.50)(20.25)1.50222sothatvar()YE=-=-=(YE)[()]Y1.50(1.00)0.50.2.WeknowfromTable2.2thatPr(Y=0)=.022,Pr(Y=1)=.078,Pr(X==.0)030,Pr(X==.1)070.So(a)m==EY()0Pr(ì=Y0)1Pr(1)+
3、ì=YY=ì.+ì.=.00221078078,m=EX()0Pr(=ìX=+ì0)1Pr(1)X=X=ì.+ì.=..00301070070(b)22sm=-EX[()]XX22=-(00.70)ìPr(XX=+-0)(10.70)ìPr(=1)22=-.(070)030030070021ì.+.ì.=.,22sm=-EY[()]YY22=-(00.78)ìPr(YY=+-0)(10.78)ìPr(=1)22=-.(078)02202207801716ì.+.ì.=..4Stock/Watson-IntroductiontoEconometri
4、cs-SecondEdition(c)Table2.2showsPr(XY===0,0)015,.Pr(XY=0,==.1)015,Pr(XY=1,==.0)007,Pr(XY===1,1)063..Sos==cov(,)XYEX[(-mm)(Y-)]XYXY==(0-0.70)(0-0.78)Pr(XY0,=0)+-.(0070)(1078)Pr(-.XY=,=01)+-.(1070)(0-.078)Pr(XY=,=10)+-.(1070)(1078)Pr(-.XY=,=11)=-.ì-.ì.+-.ì.ì.(070)(078)015(070)0
5、22015+.ì-.ì.+.030(078)007030022063ì.ì.=.0084,s0084.XYcorXY(,)===04425..ssXY02101716.ì.3.ForthetwonewrandomvariablesWX=+36andVY=207,-wehave:(a)EV()=-=-=-E(207)207()2070781454,YEYì.=.EW()(36)36()3607072=+=EX+EX=+ì.=..(b)222ss=+=var(36)X6∂=36021756,ì.=.WX222ss=-var(207)Y=(7)-∂=4
6、901716ì.=84084..VY(c)s=+-=cov(36,207)XY6(7)-covXY(,)=-420084ì.=-3528.WVs-.3528WVcorWV(,)===-04425..ssWV75684084.ì.3334.(a)E()0(Xp=ì-+ì=1)1ppkkk(b)E()0(Xp=ì-+ì=1)1pp(c)EX()0.3=22var()XE=-=(XE)[()]X0.30.09-=0.21Thus,s==0.210.46.Tocomputetheskewness,usetheformulafromexercise2.21
7、:3323E()(XE-=mXE)-3[(X)][E(XE)]+2[(X)]23=-ì+ì=0.330.320.30.084333Alternatively,EX()--m=[(10.3)ì0.3]+[(0-0.3)ì0.7]=0.084333Thus,skewness=-=EX()ms/.084/0.460=.87.SolutionstoExercisesinChapter25Tocomputethekurtosis,usetheformulafromexercise2.21:443224E()(XE-=mXE)-4[(X)][E(XE)]+6
8、[(X)][E(XE)]-3[(X)]234=-ì+ì-ì=0.340.360.330.30.0777444Alternatively,
