二元正態(tài)分布

1、 ThebivariatenormaldistributionWesaythat(X5Y)havebivariatenormaldistributioniftheirjointdensityiswhere燈、MY、aYandpareconstants.2pmx)(y_ay)MarginaldensityIf(X5Y)havebivariatenormaldensity,thenxN(“X，胃)YN0y,兄)andCorrX,Y=pIndeed,themarginaldensityofXisknowntobeZoofxY0)y)dy-ooLettingu=xrxxand9=(y燦丫)/2)(np

2、v)dvTheinnerintegralisthemeanofanNpv,(1-p2)r.v.andthussinceforaZN(0,1),CovX,Y=PXYVpv2dvE(Z)?=Va/r(Z)+(EZ)21丿OCV27TThus,CorrX,Y=p.IndependenceForr.v.sand口weknowthatifand77areindependentthenCorr,?7=0.Ontheotherhand,wemayhaveCorr,77=0evenif&耳arenotindependent.However,if（X，Y）arebivariatenormalandCorrX,Y

3、=0,thebivariatenormalpdfreducestotheproductofpdfsoftwoindependentnormalrvsX、Y.Thatis,inthiscaseCorrX,Y=0X,Yareindependent.ConditionaldensityIf（X，Y）arebivariatenormal,thentheconditionaldensitiesfyxyx）andfxY（xMareunivariatenormal.Indeed,aftersomemessyalgebrawehave（劄巧=1Thisisanormaldensitywithmean+px2x

4、andvariance只）. Example36SupposethatXN(0,1),YN(0,1)andindependent.LetU=X、V=X-Y.Showthat(t/,V)isbivariatenormal.Byindependence,wehave/x,yO,)=/exp+/forocxyoc.Sincex=uandy=vu,thentheJacobianis # #andthepdfof(/,V)isfuy(uQ)=/xy(Xq-“)x112SXP*(2+3u)2)ioThisisabivariatenormaldensitywithMU=MV=0，and并=1，ay=2,p=1/2.Generally,wehave:if(X,y)arebivariatenormal,thenforanylineartransformationU=aX+bY+V=