an introduction to probability theory - geiss

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1、AnintroductiontoprobabilitytheoryChristelGeissandStefanGeissFebruary19,20042Contents1Probabilityspaces71.1Definitionofσ-algebras......................81.2Probabilitymeasures.......................121.3Examplesofdistributions....................201.3.1Binomialdistributionwithparameter0

2、0.......211.3.3Geometricdistributionwithparameter00......................221.3.6ExponentialdistributiononRwithparameterλ>0.

3、221.3.7Poisson’sTheorem....................241.4AsetwhichisnotaBorelset..................252Randomvariables292.1Randomvariables.........................292.2Measurablemaps.........................312.3Independence...........................353Integration393.1Definitionoftheexpectedval

4、ue.................393.2Basicpropertiesoftheexpectedvalue..............423.3ConnectionstotheRiemann-integral..............483.4Changeofvariablesintheexpectedvalue............493.5Fubini’sTheorem.........................513.6Someinequalities.........................584Modesofconvergen

5、ce634.1Definitions.............................634.2Someapplications.........................6434CONTENTSIntroductionThemodernperiodofprobabilitytheoryisconnectedwithnameslikeS.N.Bernstein(1880-1968),E.Borel(1871-1956),andA.N.Kolmogorov(1903-1987).Inparticular,in1933A.N.Kolmogorovpubl

6、ishedhismodernap-proachofProbabilityTheory,includingthenotionofameasurablespaceandaprobabilityspace.Thislecturewillstartfromthisnotion,tocontinuewithrandomvariablesandbasicpartsofintegrationtheory,andtofinishwithsomefirstlimittheorems.Thelectureisbasedonamathematicalaxiomaticapproachan

7、disintendedforstudentsfrommathematics,butalsoforotherstudentswhoneedmoremathematicalbackgroundfortheirfurtherstudies.WeassumethattheintegrationwithrespecttotheRiemann-integralonthereallineisknown.Theapproach,wefollow,seemstobeinthebeginningmoredifficult.Butonceonehasasolidbasis,manythi

8、ngswillbeeas