intro to mathematical statistics hogg chapter 2

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1、Chapter2MultivariateDistributions2.1DistributionsofTwoRandomVariablesWebeginthediscussionofapairofrandomvariableswiththefollowingexample.Acoinistossedthreetimesandourinterestisintheorderednumberpair(numberofH’sonfirsttwotosses,numberofH’sonallthreetosses),whereHandTrepr

2、esent,respectively,headsandtails.LetC={TTT,TTH,THT,HTT,THH,HTH,HHT,HHH}denotethesamplespace.LetX1denotethenumberofH’sonthefirsttwotossesandX2denotethenumberofH’sonallthreeflips.Thenourinter-estcanberepresentedbythepairofrandomvariables(X1,X2).Forexample,(X1(HTH),X2(HTH))

3、representstheoutcome(1,2).Continuinginthisway,X1andX2arereal-valuedfunctionsdefinedonthesamplespaceC,whichtakeusfromthesamplespacetothespaceoforderednumberpairs.D={(0,0),(0,1),(1,1),(1,2),(2,2),(2,3)}.ThusX1andX2aretworandomvariablesdefinedonthespaceC,and,inthisexample,t

4、hespaceoftheserandomvariablesisthetwo-dimensionalsetD,whichisasubsetoftwo-dimensionalEuclideanspaceR2.Hence(X,X)isavector12functionfromCtoD.Wenowformulatethedefinitionofarandomvector.Definition2.1.1(RandomVector).GivenarandomexperimentwithasamplespaceC,considertworandomv

5、ariablesX1andX2,whichassigntoeachelementcofConeandonlyoneorderedpairofnumbersX1(c)=x1,X2(c)=x2.Thenwesaythat(X1,X2)isarandomvector.Thespaceof(X1,X2)isthesetoforderedpairsD={(x1,x2):x1=X1(c),x2=X2(c),c∈C}.WeoftendenoterandomvectorsusingvectornotationX=(X,X),where12thede

6、notesthetransposeoftherowvector(X,X).12LetDbethespaceassociatedwiththerandomvector(X1,X2).LetAbeasubsetofD.Asinthecaseofonerandomvariable,wespeakoftheeventA.WewishtodefinetheprobabilityoftheeventA,whichwedenotebyPX1,X2[A].As7374MultivariateDistributionswithrandomvariabl

7、esinSection1.5wecanuniquelydefinePX1,X2intermsofthecumulativedistributionfunction(cdf),whichisgivenbyFX1,X2(x1,x2)=P[{X1≤x1}∩{X2≤x2}],(2.1.1)forall(x,x)∈R2.BecauseXandXarerandomvariables,eachoftheevents1212intheaboveintersectionandtheintersectionoftheeventsareeventsinth

8、eoriginalsamplespaceC.Thustheexpressioniswelldefined.Aswithrandomvariables,wewriteP[{X1≤x1}∩{X2≤x2}]asP[X1≤x1,X2≤x2].A