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WernerLinde ProbabilityTheory DeGruyterGraduate Also of interest ProbabilityTheoryandStatisticalApplications.AProfoundTreatisefor Self-Study PeterZörnig,2016 ISBN978-3-11-036319-7,e-ISBN(PDF)978-3-11-040271-1, e-ISBN(EPUB)978-3-11-040283-4 StochasticFinance:AnIntroductioninDiscreteTime HansFöllmer,AlexanderSchied,4thEdition,2016 ISBN978-3-11-046344-6,e-ISBN(PDF)978-3-11-046345-3, e-ISBN(EPUB)978-3-11-046346-0 AsymptoticStatistics:WithaViewtoStochasticProcesses ReinhardHöpfner,2014 ISBN978-3-11-025024-4,e-ISBN(PDF)978-3-11-025028-2, e-ISBN(EPUB)978-3-11-036778-2 BrownianMotion:AnIntroductiontoStochasticProcesses RenéL.Schilling,LotharPartzsch,2ndEdition,2014 ISBN978-3-11-030729-0,e-ISBN(PDF)978-3-11-030730-6, e-ISBN(EPUB)978-3-11-037398-1 Stochastics:IntroductiontoProbabilityandStatistics Hans-OttoGeorgii,2ndEdition,2012 ISBN978-3-11-029254-1,e-ISBN(PDF)978-3-11-029360-9 Werner Linde Probability Theory A First Course in Probability Theory and Statistics MathematicsSubjectClassification2010 Primary:60-01,62-01;Secondary:60A05 Author Prof.Dr.WernerLinde Friedrich-Schiller-UniversitätJena FakultätfürMathematik&Informatik InstitutfürStochastik Prof.fürStochastischeAnalysis D-07737Jena [email protected] and UniversityofDelaware DepartmentofMathematicalSciences 501EwingHall NewarkDE,19716 [email protected] ISBN978-3-11-046617-1 e-ISBN(PDF)978-3-11-046619-5 e-ISBN(EPUB)978-3-11-046625-6 LibraryofCongressCataloging-in-PublicationData ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress. BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;detailed bibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2016WalterdeGruyterGmbH,Berlin/Boston Typesetting:IntegraSoftwareServicesPvt.Ltd. Printingandbinding:CPIbooksGmbH,Leck Coverimage:WernerLinde (cid:2)8Printedonacid-freepaper PrintedinGermany www.degruyter.com TomywifeKarin Preface Thisbookisintendedasanintroductorycourseforstudentsinmathematics,phys- ical sciences, engineering, or in other related fields. It is based on the experience ofprobabilitylecturestaughtduringthepast25years,wherethespectrumreached fromtwo-hourintroductorycourses,overMeasureTheoryandadvancedprobability classes,tosuchtopicsasStochasticProcessesandMathematicalStatistics.Until2012 theselecturesweredeliveredtostudentsattheUniversityofJena(Germany),andsince 2013tothoseattheUniversityofDelawareinNewark(USA). ThebookisthecompletelyrevisedversionoftheGermanedition“Stochastikfür dasLehramt,”whichappearedin2014atDeGruyter.AtmostuniversitiesinGermany, there exist special classes in Probability Theory for students who want to become teachersofmathematicsinhighschools.BesidesbasicfactsaboutProbabilityTheory, thesecoursesarealsosupposedtogiveanintroductionintoMathematicalStatistics. Thus,theoriginalmainintentionfortheGermanversionwastowriteabookthathelps thosestudentsunderstandProbabilityTheorybetter.Butsoonthebookturnedoutto alsobeusefulasintroductionforstudentsinotherfields,e.g.inmathematics,phys- ics,andsoon.Thuswedecided,inordertomakethebookapplicableforabroader audience,toprovideatranslationintheEnglishlanguage. DuringnumerousyearsofteachingIlearnedthefollowing: – Probabilisticquestionsareusuallyeasytoformulate,generallyhaveatightrela- tiontoeverydayproblems,andthereforeattracttheinterestoftheaudience.Every studentknowsthephenomenathatoccurwhenonerollsadie,playscards,tosses acoin,orplaysalottery.Thus,aninitialinterestinProbabilityTheoryexists. – In contrast, after a short time many students have very serious difficulties with understanding the presented topics. Consequently, a common opinion among students is that Probability Theory is a very complicated topic, causing a lot of problemsandtroubles. SurelythereexistseveralreasonsforthebadimageofProbabilityTheoryamongstu- dents.But,aswebelieve,themostimportantoneisasfollows.InProbabilityTheory, thetypeofproblemsandquestionsconsidered,aswellasthewayofthinking,differs considerablyfromtheproblems,questions,andthinkinginotherfieldsofmathem- atics,i.e.,fromfieldswithwhichthestudentsbecameacquaintedbeforeattendinga probabilitycourse.Forexample,inCalculusafunctionhasawell-describeddomain ofdefinition;mostlyitisdefinedbyaconcreteformula,hascertainpropertiesascon- tinuity,differentiability,andsoon.Afunctionissomethingveryconcretewhichcan bemadevividbydrawingitsgraph.Incontrast,inProbabilityTheoryfunctionsare mostlyinvestigatedasrandomvariables.Theyaredefinedonacompletelyunimport- ant,nonspecifiedsamplespace,andtheygenerallydonotpossessaconcreteformula fortheirdefinition.Itmayevenhappenthatonlytheexistenceofafunction(random variable) is known. The only property of a random variable which really matters is VIII Preface thedistributionofitsvalues.Thisandmanyothersimilartechniquesmakethewhole theorysomethingmysteriousandnotcompletelycomprehensible. Consideringthisobservation,weorganizedthebookinawaythattriestomake probabilistic problems more understandable and that puts the focus more onto ex- planationsofthedefinitions,notations,andresults.Thetoolsweusetodothisare examples;wepresentatleastonebeforeanewdefinition,inordertomotivateit,fol- lowedbymoreexamplesafterthedefinitiontomakeitcomprehensible.Hereweact uponthemaximexpressedbyEinstein’squote1: Exampleisn’tanotherwaytoteach,itistheonlywaytoteach. PresentingthebasicresultsandmethodsinProbabilityTheorywithoutusingresults, facts,andnotationsfromMeasureTheoryis,inouropinion,asdifficultastosquare the circle. Either one restricts oneself to discrete probability measures and random variables or one has to be unprecise. There is no other choice! In some places, it is possibletoavoidtheuseofmeasuretheoreticfacts,suchastheLebesgueintegral,or theexistenceofinfiniteproductmeasures,andsoon,butthepriceishigh.2Ofcourse, IalsostruggledwiththeproblemofmissingfactsfromMeasureTheorywhilewriting this book. Therefore, I tried to include some ideas and some results about 3-fields, measures, and integrals, hoping that a few readers become interested and want to learnmoreaboutMeasureTheory.Forthose,werefertothebooks[Coh13],[Dud02], or[Bil12]asgoodsources. Inthiscontext,letusmakesomeremarkabouttheverificationofthepresented results. Whenever it was possible, we tried to prove the stated results. Times have changed;whenIwasastudent,everytheorempresentedinamathematicallecture was proved – really every one. Facts and results without proof were doubtful and soon forgotten. And a tricky and elegant proof is sometimes more impressive than theprovenresult(atleasttous).Hopefully,somereaderswilllikesomeoftheproofs inthisbookasmuchaswedid. One of most used applications of Probability Theory is Mathematical Statistics. WhenImetformerstudentsofmine,Ioftenaskedthemwhichkindofmathematics they are mainly using now in their daily work. The overwhelming majority of them answeredthatoneoftheirmainfieldsofmathematicalworkisstatisticalproblems. Therefore,wedecidedtoincludeanintroductorychapteraboutMathematicalStatist- ics. Nowadays, due to the existence of good and fast statistical programs, it is very easy to analyze data, to evaluate confidence regions, or to test a given hypotheses. Butdothosewhousetheseprogramsalsoalwaysknowwhattheyaredoing?Since 1 Seehttp://www.alberteinsteinsite.com/quotes/einsteinquotes.html 2 Forexample,severalyearsago,toavoidtheuseoftheLebesgueintegral,Iintroducedtheexpected valueofarandomvariableasaRiemannintegralviaitsdistributionfunction.Thisismathematically correct,butattheendalmostnostudentsunderstoodwhattheexpectedvaluereallyis.Trytoprove thattheexpectedvalueislinearusingthisapproach! Preface IX wedoubtthatthisisso,westressedthefocusinthischaptertothequestionofwhy themainstatisticalmethodsworkandonwhatmathematicalbackgroundtheyrest. Wealsoinvestigatehowprecisestatisticaldecisionsareandwhatkindsoferrorsmay occur. The organization of this book differs a little bit from those in many other first- course books about Probability Theory. Having Measure Theory in the back of our mindscausesustothinkthatprobabilitymeasuresarethemostimportantingredi- entofProbabilityTheory;randomvariablescomeinsecond.Onthecontrary,many otherauthorsgoexactlytheotherway.Theystartwithrandomvariables,andprob- ability measures then occur as their distribution on their range spaces (mostly R). Inthiscase,astandardnormalprobabilitymeasuredoesnotexist,onlyastandard normaldistributedrandomvariable.Bothapproacheshavetheiradvantagesanddis- advantages,butaswesaid,forustheprobabilitymeasuresareinterestingintheirown right,andthereforewestartwiththeminChapter1,followedbyrandomvariablesin Section3. Thebookalsocontainssomefactsandresultsthataremoreadvancedandusually notpartofanintroductorycourseinProbabilityTheory.Suchtopicsare,forexample, theinvestigationofproductmeasures,orderstatistics,andsoon.Wehaveassigned those more involved sections with a star. They may be skipped at a first reading withoutlossinthefollowingchapters. Attheendofeachchapter,onefindsacollectionofsomeproblemsrelatedtothe contentsofthesection.Herewerestrictedourselvestoafewproblemsintheactual task;thesolutionsoftheseproblemsarehelpfultotheunderstandingofthepresented topics.Theproblemsaremainlytakenfromourcollectionofhomeworksandexams during the past years. For those who want to work with more problems we refer to manybooks,ase.g.[GS01a],[Gha05],[Pao06],or[Ros14],whichcontainahugecollec- tionofprobabilisticproblems,rangingfromeasytodifficult,fromnaturaltoartificial, frominterestingtoboring. FinallyIwanttoexpressmythankstothosewhosupportedmyworkatthetrans- lationandrevisionofthepresentbook.ManystudentsattheUniversityofDelaware helped me to improve my English and to correct wrong phrases and wrong expres- sions. To mention all of them is impossible. But among them were a few students who read whole chapters and, without them, the book would have never been fin- ished(orreadable).InparticularIwanttomentionEmilyWagnerandSpencerWalker. They both did really a great job. Many thanks! Let me also express my gratitude to ColleenMcInerney,RachelAustin,DanielAtadan,andQuentinDubroff,allstudents inDelawareandattendingmyclassesforsometime.Theyalsoreadwholesectionsof thebookandcorrectedmybrokenEnglish.Finally,mythanksgotoProfessorAnne LeuchtfromtheTechnicalUniversityinBraunschweig(Germany);herfieldofworkis MathematicalStatistics,andherhintsandremarksaboutChapter8inthisbookwere importanttome.