資源描述：
《Mathematical Statistics - Jun Shaochap1》由會員上傳分享，免費在線閱讀，更多相關內(nèi)容在學術論文-天天文庫。
1、Chapter1ProbabilityTheoryMathematicalstatisticsreliesonprobabilitytheory,whichinturnisbasedonmeasuretheory.Thepresentchapterprovidessomeprincipalconceptsandnotationalconventionsofprobabilitytheory,andsomeimportantre-sultsthatareusefultoolsinstatistics.Amorecompleteaccountofproba-bilitytheoryc
2、anbefoundinastandardtextbook,forexample,Billingsley(1986),Chung(1974),orLo`eve(1977).Thereaderisassumedtobefamiliarwithsetoperationsandsetfunctions(mappings)inadvancedcalculus.1.1ProbabilitySpacesandRandomElementsInanelementaryprobabilitycourse,onede?nesarandomexperimenttobeanexperimentwhoseo
3、utcomecannotbepredictedwithcertainty,andtheprobabilityofA(acollectionofpossibleoutcomes)tobethefractionoftimesthattheoutcomeoftherandomexperimentresultsinAinalargenumberoftrialsoftherandomexperiment.Arigorousandlogicallyconsistentde?nitionofprobabilitywasgivenbyA.N.Kolmogorovinhismeasure-theo
4、reticfundamentaldevelopmentofprobabilitytheoryin1933(Kolmogorov,1933).1.1.1σ-?eldsandmeasuresLet?beasetofelementsofinterest.Forexample,?canbeasetofnumbers,asubintervaloftherealline,orallpossibleoutcomesofarandomexperiment.Inprobabilitytheory,?isoftencalledtheoutcomespace,whereasinstatisticalt
5、heory,?iscalledthesamplespace.Thisisbecauseinprobabilityandstatistics,?isusuallythesetofallpossibleoutcomesofarandomexperimentunderstudy.121.ProbabilityTheoryAmeasureisanaturalmathematicalextensionofthelength,area,orvolumeofsubsetsintheone-,two-,orthree-dimensionalEuclideanspace.Inagivensampl
6、espace?,ameasureisasetfunctionde?nedforcertainsubsetsof?.Itisnecessaryforthiscollectionofsubsetstosatisfycertainproperties,whicharegiveninthefollowingde?nition.De?nition1.1.LetFbeacollectionofsubsetsofasamplespace?.Fiscalledaσ-?eld(orσ-algebra)ifandonlyifithasthefollowingproperties.(i)Theempt
7、yset?∈F.(ii)IfA∈F,thenthecomplementAc∈F.(iii)IfAi∈F,i=1,2,...,thentheirunion∪Ai∈F.Apair(?,F)consistingofaset?andaσ-?eldFofsubsetsof?iscalledameasurablespace.TheelementsofFarecalledmeasurablesetsinmeasuretheoryoreventsinprobabilityandstatistic
