Chapters 5. Multivariate Probability Distributions

1 , ,Xi=1ifthei-thtossyieldsheads, ,X2,.., ,andexpressthetotalnumberofheadsintermso fX1,X2,.., ,say(G,Y).Independence? X=X1+X2+ + (X,Y)canbedescribedbythejointprobability function{pij} (X=xi,Y=yj).Weshouldhavepij 0and i jpij=1. (X,Y)canbede-scribedviaanonnegativejoint densityfunctionf(x,y)suchthatforanysubse tA R2,P((X,Y) A)= Af(x,y) R2f(x,y)dxdy= (cdf)forarandomvector(X,Y)isdefinedasF(x ,y).=P(X x,Y y)forx,y :F(x,y)= xi x yj yP(X=xi,Y=yj) :F(x,y)= x y f(u,v)dudvf(x,y)= 2F(x,y) x (X,Y)hasadensityf(x,y)={cxye (x+y),ifx>0,y>00,otherwiseDeterminetheva lueofcandcalculateP(X+Y 1). ,30%areRepublicans,50%areDemocrates, {1,ifRepublican0,otherwiseY={1,ifDemocra t0, (X,Y)isthecoordinatesofarandomlyselected pointfromthedisk{(x,y): x2+y2 2}.Findthejointdensityof(X,Y).CalcualteP (X<Y)andtheprobabilitythat(X,Y)isintheun itdisk{(x,y): x2+y2 1}.MarginalDistributionsConsiderarandomv ector(X,Y).}}}

2 :ThemarginaldistributionforXisgivenbyP(X =xi)= jP(X=xi,Y=yj)= :ThemarginaldensityfunctionforXisgivenby fX(x).= Rf(x,y) :ThemarginalcdfforXisFX(x)=F(x, ). (x,y)={e x,0<y<x0, :ConditionaldistributionofYgivenX=xicanb edescribedbyP(Y=yj|X=xi)=P(X=xi,Y=yj)P(X =xi)= :ConditionaldensityfunctionofYgivenX=xis definedbyf(y|x).=f(x,y)fX(x)= (Y a|X=x)andconditionalprobabilitysuchasP(Y a|X x).Forthelatter,onecanusetheusualdefinit ionofconditionalprobabilityandP(Y a|X x)=P(X x,Y a)P(X x)Butfortheformer,thisisnotvalidanymores inceP(X=x)= (Y a|X=x)= a f(y|x) {Bi} (A)= iP(A|Bi)P(Bi) R2,wehaveP((X,Y) G)= iP((xi,Y) G|X=xi)P(X=xi) R2,wehaveP((X,Y) G)= RP((x,Y) G|X=x)fX(x) ,findthedistribution,theexpectedvalue, (X,Y)withjointdensityf(x,y)={e x,0<y<x0,otherwiseCompute(a)P(X Y 2|X 10),(b)P(X Y 2|X=10),(c)GivenX=10, ,thedistributionofXisPoissonwithparamete r . :TworandomvariablesX,Yaresaidtobeindepen dentifforanysubsetsA,B RP(X A,Y B)=P(X A)P(Y B)Definition2:TworandomvariablesX, ,YarediscreteP(X=xi,Y=yj)=P(X=xi)P(Y=yj) .}}

3 ,Yarecontinuousf(x,y)=fX(x)fY(y).Remark: Independenceifandonlyifconditionaldistri bution :SupposeX, ,g(X)andh(Y)arealsoindependentRemark:Two continuousrandomvariablesareindependenti fandonlyifitsdensityf(x,y)canbewrittenin split-formoff(x,y)=g(x)h(y). ! ,withfasthejointdensity? (x,y)={(x+y),0<x<1,0<y<10, (x,y)={6x2y,0 x 1,0 y 10, (x,y)={8xy,0 y x 10, ,X2,..,Xnareindependentidenticallydistri buted(iid)BernoullirandomvariableswithP( Xi=1)=p,P(Xi=0)=1 +X2+ + (n;p)andYisdistributedasB(m;p).IfXandYar eindependent,whatisthedistributionofX+Y? ,Yareindependentrandomvariableswithdistr ibutionP(X=k)=P(Y=k)=15,k=1,2,..,5 FindP(X+Y 5). andYisexponentiallydistributedwithrate .FindoutthejointdensityofXandYandcompute P(X<Y). and +Y?Expectedvalues Discreterandomvector(X,Y):E[g(X,Y)]= i jg(xi,yj)P(X=xi,Y=yj). Continuousrandomvector(X,Y):E[g(X,Y)]= R2g(x,y)f(x,y) ,Yandanyconstantsa,b,E[aX+bY]=aE[X]+bE[Y ].}}}

4 [XY]=E[X]E[Y]. {1,ifthei-thcardisanAce0,otherwise(a)Are X1,..,X4independent?(b)AreX1,..,X4identi callydistributed?(c) , (x,y)= 2e x,0 y x< , ,covariance,andcorrelationTworandomvaria blesX,Ywithmean X, (X,Y).=E[(X X)(Y Y)].Let Xand .=Cov(X,Y) X Y Whatdoescorrelationmean?[(height,weight) ,(houseage,houseprice)] Thecorrelationcoefficientsatisfies 1 1. Var[X]=Cov(X,X). ,Cov(X,Y)=E[XY] E[X]E[Y]. ,wehaveCov(X,Y)= ,b,c,dandrandomvariablesX,Y,Cov(aX+b,cY+ d)=acCov(X,Y). (X,Y)=Cov(Y,X) (X,Y+Z)=Cov(X,Y)+Cov(X,Z). ,Cov(X,a)= [X+Y]=Var[X]+Var[Y]+2 Cov(X,Y) [X+Y]=Var[X]+Var[Y]. ,b,wehaveVar[aX+bY]=a2 Var[X]+b2 Var[Y]+2abCov(X,Y). ,andE[X]=1,E[Y]= (i)E[2X Y+3];(ii)Var[2X Y+3];(iii)E[XY];(iv)E[X2Y2];(v)Cov(X,XY) . (n;p).3.(Estimateof )LetX1,X2,.., X=X1+X2+ + {(x,y):x2+y2 1}.AreXandYindependent?Aretheyuncorrelat ed( =0)?5.(Revisitthe4-cardexample). , ,withp1+p2+ +pk= (Y1.)}

5 ,Yk).NotethatY1+Y2+..+Yk= ,..,yksuchthaty1+ +yk=n,P(Y1=y1,..,Yk=yk)=(ny1y2 yk)py11 [0,1]. (Y1,Y2,..,Yk)ismultinomialwithparameters (n;p1,p2,..,pk).(a)Whatisthedistribution of,sayY1?(b)Whatisthejointdistributionof ,say(Y1,Y2)? ,14%sophomores,38%juniors,and32% (Y1,Y2,..,Yk)hasmultinomialdistributionw ithparameters(n;p1,p2,..,pk). [Yi]=npi,Var[Yi]=npi(1 pi) (Yi,Yj)= npipj,i6= [h(Y)|X=x] :E[h(Y)|X=xi]= jh(yj)P(Y=yj|X=xi) :E[h(Y)|X=x]= Rh(y)f(y|x) ,E[aY+bZ|X=x]=aE[Y|X=x]+bE[Z|X=x] ,thenforanyx,E[Y|X=x] E[Y] [Y|X]isarandomvariable!Explanation:E[Y|X =x]canbeviewedasafunctionofx WriteE[Y|X=x]=g(x) E[Y|X]=g(X) :ForanyrandomvariablesXandYE[E[Y|X]]=E[Y ]..Furtherassumethat variesweekfromweekandisassumedtobeexpone ntialdistributedwithrate . ,sayA,B, ,theparticlehas50% ,thenonaveragehowmanystepsitneedstoarriv eatlocationB?