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1、Chapter1ProbabilityTheoryMathematicalstatisticsreliesonprobabilitytheory,whichinturnisbasedonmeasuretheory.Thepresentchapterprovidessomeprincipalconceptsandnotationalconventionsofprobabilitytheory,andsomeimportantre-sultsthatareusefultoolsinstatistics.Amorecompleteaccountofproba-bilitytheoryc
2、anbefoundinastandardtextbook,forexample,Billingsley(1986),Chung(1974),orLo`eve(1977).Thereaderisassumedtobefamiliarwithsetoperationsandsetfunctions(mappings)inadvancedcalculus.1.1ProbabilitySpacesandRandomElementsInanelementaryprobabilitycourse,onedefinesarandomexperimenttobeanexperimentwhoseo
3、utcomecannotbepredictedwithcertainty,andtheprobabilityofA(acollectionofpossibleoutcomes)tobethefractionoftimesthattheoutcomeoftherandomexperimentresultsinAinalargenumberoftrialsoftherandomexperiment.ArigorousandlogicallyconsistentdefinitionofprobabilitywasgivenbyA.N.Kolmogorovinhismeasure-theo
4、reticfundamentaldevelopmentofprobabilitytheoryin1933(Kolmogorov,1933).1.1.1σ-fieldsandmeasuresLetΩbeasetofelementsofinterest.Forexample,Ωcanbeasetofnumbers,asubintervaloftherealline,orallpossibleoutcomesofarandomexperiment.Inprobabilitytheory,Ωisoftencalledtheoutcomespace,whereasinstatisticalt
5、heory,Ωiscalledthesamplespace.Thisisbecauseinprobabilityandstatistics,Ωisusuallythesetofallpossibleoutcomesofarandomexperimentunderstudy.121.ProbabilityTheoryAmeasureisanaturalmathematicalextensionofthelength,area,orvolumeofsubsetsintheone-,two-,orthree-dimensionalEuclideanspace.Inagivensampl
6、espaceΩ,ameasureisasetfunctiondefinedforcertainsubsetsofΩ.Itisnecessaryforthiscollectionofsubsetstosatisfycertainproperties,whicharegiveninthefollowingdefinition.Definition1.1.LetFbeacollectionofsubsetsofasamplespaceΩ.Fiscalledaσ-field(orσ-algebra)ifandonlyifithasthefollowingproperties.(i)Theempt
7、yset∅∈F.(ii)IfA∈F,thenthecomplementAc∈F.(iii)IfAi∈F,i=1,2,...,thentheirunion∪Ai∈F.Apair(Ω,F)consistingofasetΩandaσ-fieldFofsubsetsofΩiscalledameasurablespace.TheelementsofFarecalledmeasurablesetsinmeasuretheoryoreventsinprobabilityandstatistic
