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1、AnintroductiontoprobabilitytheoryChristelGeissandStefanGeissFebruary19,20042Contents1Probabilityspaces71.1De?nitionofσ-algebras......................81.2Probabilitymeasures.......................121.3Examplesofdistributions....................201.3.1Binomialdistribution
2、withparameter00.......211.3.3Geometricdistributionwithparameter00......................221.3.6Exponentialdist
3、ributiononRwithparameterλ>0.221.3.7Poisson’sTheorem....................241.4AsetwhichisnotaBorelset..................252Randomvariables292.1Randomvariables.........................292.2Measurablemaps.........................312.3Independence...........................35
4、3Integration393.1De?nitionoftheexpectedvalue.................393.2Basicpropertiesoftheexpectedvalue..............423.3ConnectionstotheRiemann-integral..............483.4Changeofvariablesintheexpectedvalue............493.5Fubini’sTheorem.........................513.6Some
5、inequalities.........................584Modesofconvergence634.1De?nitions.............................634.2Someapplications.........................6434CONTENTSIntroductionThemodernperiodofprobabilitytheoryisconnectedwithnameslikeS.N.Bernstein(1880-1968),E.Borel(1871-19
6、56),andA.N.Kolmogorov(1903-1987).Inparticular,in1933A.N.Kolmogorovpublishedhismodernap-proachofProbabilityTheory,includingthenotionofameasurablespaceandaprobabilityspace.Thislecturewillstartfromthisnotion,tocontinuewithrandomvariablesandbasicpartsofintegrationtheory,and
7、to?nishwithsome?rstlimittheorems.Thelectureisbasedonamathematicalaxiomaticapproachandisintendedforstudentsfrommathematics,butalsoforotherstudentswhoneedmoremathematicalbackgroundfortheirfurtherstudies.WeassumethattheintegrationwithrespecttotheRiemann-integralontherealli
8、neisknown.Theapproach,wefollow,seemstobeinthebeginningmoredi?cult.Butonceonehasasolidbasis,manythingswillbeeas
