AN_INTRODUCTION_TO_PROBABIL.PDF

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Anintroductiontoprobabilitytheory
ChristelGeissandStefanGeiss
February19,2004
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Contents
1Probabilityspaces 7
1.1Definitionof-algebras...................... 8
1.2Probabilitymeasures.......................12
1.3Examplesofdistributions....................20
1.3.1Binomialdistributionwithparameter0< p <1....20
1.3.2Poissondistributionwithparameter >0.......21
1.3.3Geometricdistributionwithparameter0< p <1...21
1.3.4Lebesguemeasureanduniformdistribution......21
1.3.5GaussiandistributiononRwithmeanm2Rand
variance 2 >0......................22
1.3.6ExponentialdistributiononRwithparameter >0.22
1.3.7Poisson’sTheorem....................24
1.4AsetwhichisnotaBorelset..................25
2Randomvariables 29
2.1Randomvariables.........................29
2.2Measurablemaps.........................31
2.3Independence...........................35
3Integration 39
3.1Definitionoftheexpectedvalue.................39
3.2Basicpropertiesoftheexpectedvalue..............42
3.3ConnectionstotheRiemann-integral..............48
3.4Changeofvariablesintheexpectedvalue............49
3.5Fubini’sTheorem.........................51
3.6Someinequalities.........................58
4Modesofconvergence 63
4.1Definitions.............................63
4.2Someapplications.........................64
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4 CONTENTS
Introduction
ThemodernperiodofprobabilitytheoryisconnectedwithnameslikeS.N.
Bernstein(1880-1968),E.Borel(1871-1956),andA.N.Kolmogorov(1903-
1987).Inparticular,in1933A.N.Kolmogorovpublishedhismodernap-
proachofProbabilityTheory,includingthenotionofameasurablespace
andaprobabilityspace.Thislecturewillstartfromthisnotion,tocontinue
withrandomvariablesandbasicpartsofintegrationtheory,andtofinish
withsomefirstlimittheorems.
Thelectureisbasedonamathematicalaxiomaticapproachandisintended
forstudentsfrommathematics,butalsoforotherstudentswhoneedmore
mathematicalbackgroundfortheirfurtherstudies.Weassumethatthe
integrationwithrespecttotheRiemann-integralonthereallineisknown.
Theapproach,wefollow,seemstobeinthebeginningmoredicult.But
onceonehasasolidbasis,manythingswillbeeasierandmoretransparent
later.Letusstartwithanintroducingexampleleadingustoaproblem
whichshouldmotivateouraxiomaticapproach.
Example.Wewouldliketomeasurethetemperatureoutsideourhome.
Wecandothisbyanelectronicthermometerwhichconsistsofasensor
outsideandadisplay,includingsomeelectronics,inside.Thenumberweget
fromthesystemisnotcorrectbecauseofseveralreasons.Forinstance,the
calibrationofthethermometermightnotbecorrect,thequalityofthepower-
supplyandtheinsidetemperaturemighthavesomeimpactontheelectronics.
Itisimpossibletodescribeallthesesourcesofuncertaintyexplicitly.Hence
oneisusingprobability.Whatistheidea?
LetusdenotetheexacttemperaturebyTandthedisplayedtemperature
byS,sothatthedierenceT−Sisinfluencedbytheabovesourcesof
uncertainty.Ifwewouldmeasuresimultaneously,byusingthermometersof
thesametype,wewouldgetvaluesS 1 ,S 2 ,...withcorrespondingdierences
D 1 :=T−S 1 , D 2 :=T−S 2 , D 3 :=T−S 3 ,...
Intuitively,wegetrandomnumbersD 1 ,D 2 ,...havingacertaindistribution.
Howtodevelopanexactmathematicaltheoryoutofthis?
Firstly,wetakeanabstractset.Eachelement!2willstandfora
specificconfigurationofouroutersourcesinfluencingthemeasuredvalue.
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