数学专业英语12.docx

数学专业英语12.docx

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数学专业英语12

MathematicalEnglish

Dr.XiaominZhang

Email:

zhangxiaomin@

§2.12ProbabilityTheoryandMathematicalStatistics

TEXTASpecialterminologypeculiartoprobabilitytheory

Indiscussionsinvolvingprobability,oneoftenseesphrasesfromeverydaylanguagesuchas“twoeventsareequallylikely,”“aneventisimpossible,”or“aneventiscertaintooccur.”Expressionsofthissorthaveintuitiveappealanditisbothpleasantandhelpfultobeabletoemploysuchcolorfullanguageinmathematicaldiscussions.Beforewecandoso,however,itisnecessarytoexplainthemeaningofthislanguageintermsofthefundamentalconceptsofourtheory.

Becauseofthewayprobabilityisusedinpractice,itisconvenienttoimaginethateachprobabilityspace（S,B,P）isassociatedwitharealorconceptualexperiment.TheuniversalsetScanthenbethoughtofasthecollectionofallconceivableoutcomesoftheexperiment,asintheexampleofcointossingdiscussedintheforegoingsection.EachelementofSiscalledanoutcomeorasampleandthesubsetsofSthatoccurintheBooleanalgebraBarecalledevents.Thereasonsforthisterminologywillbecomemoreapparentwhenwetreatsomeexamples.

Assumewehaveaprobabilityspace（S,B,P）associatedwithanexperiment.LetAbeanevent,andsupposetheexperimentisperformedandthatitsoutcomeisx.（Inotherwords,letxbeapointofS.）ThisoutcomexmayormaynotbelongtothesetA.Ifitdoes,wesaythattheeventAhasoccurred.Otherwise,wesaythattheeventAhasnotoccurred,inwhichcasexA',sothecomplementaryeventA'hasoccurred.AneventAiscalledimpossibleifA=,becauseinthiscasenooutcomeoftheexperimentcanbeanelementofA.TheeventAissaidtobecertainifA=S,becausetheneveryoutcomeisautomaticallyanelementofA.

EacheventAhasaprobabilityP（A）assignedtoitbytheprobabilityfunctionP.（TheactualvalueofP（A）orthemannerinwhichP（A）isassignedisnotconcernusatpresent.）ThenumberP（A）isalsocalledtheprobabilitythatanoutcomeoftheexperimentisoneoftheelementsofA.WealsosaythatP（A）istheprobabilitythattheeventAoccurswhentheexperimentisperformed.

TheimpossibleeventmustbeassignedprobabilityzerobecausePisfinitelyadditivemeasure.However,theremaybeeventswithprobabilityzerothatarenotimpossible.Inotherwords,someofthenonemptysubsetsofSmaybeassignedprobabilityzero.ThecertaineventSmustbeassignedprobability1bytheverydefinitionofprobability,buttheremaybeothersubsetsaswellthatareassignedprobability1.Inexample1ofSection6.8therearenonemptysubsetswithprobabilityzeroandpropersubsetsofSthathaveprobability1.

TwoeventsAandBaresaidtobeequallylikelyifP（A）=P（B）.TheeventAiscalledmorelikelythanBifP（A）>P（B）,andatleastaslikelyasBifP（A）P（B）.Table2-12-1providesaglossaryorfurthereverydaylanguagethatisoftenusedinprobabilitydiscussions.ThelettersAandBrepresentevents,andxrepresentsanoutcomeofanexperimentassociatedwiththesamplespaceS.Eachentryintheleft-handcolumnisastatementabouttheeventsAandB,andthecorrespondingentryintheright-handcolumndefinesthestatementintermsofsettheory.

Notations

probabilityfunctionherethevalueofprobabilityfunctionPatpointAistheprobabilitythattheeventAoccurs.Generally,TheprobabilityfunctionP（x）（alsocalledtheprobabilitydensityfunctionordensityfunction）ofacontinuousdistributionisdefinedasthederivativeofthe（cumulative）distributionfunctionD（x）,

so

Aprobabilityfunctionsatisfies

andisconstrainedbythenormalizationcondition,

Specialcasesare

Tofindtheprobabilityfunctioninasetoftransformedvariables,findtheJacobian.Forexample,Ifu=u（x）,then

so

Similarly,ifu=u（x,y）andv=v（x,y）,then

GivennprobabilityfunctionsP1（x）,P2（x）,...,Pn（x）,thesumdistributionX+Y+…+Zhasprobabilityfunction

where（x）isadeltafunction.Similarly,theprobabilityfunctionforthedistributionofXY…Zisgivenby

ThedifferencedistributionX-Yhasprobabilityfunction

andtheratiodistributionX/Yhasprobabilityfunction

TEXTBtwobasicstatisticsconcepts—populationandsample

Intheprecedingsections,wecitedafewexamplesofsituationswhereevaluationoffactualinformationisessentialforacquiringnewknowledge.Althoughtheseexamplesaredrawnfromwidelydifferingfieldsandonlysketchydescriptionsofthescopeandobjectivesofthestudiesareprovided,afewcommoncharacteristicsarereadilydiscernible.

First,inordertoacquirenewknowledge,relevantdatemustbecollected.Second,someamountofvariabilityinthedataisunavoidableeventhoughobservationsaremadeunderthesameorcloselysimilarconditions.Forinstance,thetreatmentforanallergymayprovidelong-lastingreliefforsomeindividualswhereasitmaybringonlytransientrelieforevennoneatalltoothers.Likewise,itisunrealistictoexpectthatcollegefreshmenwhosehighschoolrecordswerealikewouldperformequallywellincollege.Naturedoesnotfollowsucharigidlaw.

Athirdnotablefeatureisthataccesstoacompletesetofdataiseitherphysicallyimpossibleorfromapracticalstandpointnotfeasible.Whendataareobtainedfromlaboratoryexperimentsorfieldtrials,nomatterhowmuchexperimentationhasbeenperformed,morecanalwaysbedone.Inpublicopinionorconsumerexpenditurestudies,acompletebodyofinformationwouldemergeonlyifdataweregatheredfromeveryindividualinthenation—undoubtedlyamonumentalifnotimpossibletask.Tocollectanexhaustivesetofdatarelatedtothedamagesustainedbyallcarsofaparticularmodelundercollisionataspecifiedspeed,everycarofthatmodelcomingofftheproductionlineswouldhavetobesubjectedtoacollision!

Thus,thelimitationsoftime,resources,andfacilities,andsometimesthedestructivenatureofthetesting,meanthatwemustworkwithincompleteinformation—thedatathatareactuallycollectedinthecourseofanexperimentalstudy.

Theprecedingdiscussionshighlightadistinctionbetweenthedatasetthatisactuallyacquiredthroughtheprocessofobservationandthevastcollectionofallpotentialobservationthatcanbeconceivedingivencontext.Thestatisticalnamefortheformerissample;forthelatter,itispopulation,orstatisticalpopulation.Tofurtherelucidatetheseconcepts,weobservethateachmeasurementinthedataasoriginatesfromadistinctsourcewhichmaybeapatient,tree,farm,household,orsomeotherentitydependingontheobjectofastudy.Thesourceofeachmeasurementiscalledasamplingunit,orsimply,aunit.Asampleorsampledatasetthenconsistsofmeasurementsrecordedforthoseunitsthatareactuallyobserved.Theobservedunitsconstituteapartofafarlargercollectionaboutwhichwewishtomakeinferences.Thesetofmeasurementsthatwouldresultofalltheunitsinthelargercollectioncouldbeobservedisdefinedasthepopulation.

Definition1Astatisticalpopulationisthesetofmeasurements（orrecordofsomequalitativetrait）correspondingtotheentirecollectionofunitsaboutwhichinformationissought.

Thepopulationrepresentsthetargetofaninvestigation.Welearnaboutthepopulationbytakingasamplefromthepopulation.

Definition2Asamplefromastatisticalpopulationisthesetofmeasurementsthatareactuallycollectedinthecourseofaninvestigation.

SUPPLEMENTABertrand'sparadox

Consideranequilateraltriangleinscribedinacircle.Supposeachordofthecircleischosenatrandom.Whatistheprobabilitythatthechordislongerthanasideofthetriangle?

ThisproblemwasoriginallyposedbyJosephBertrandinhiswork,Calculdesprobabilités（1888）.Bertrandgavethreearguments,allapparentlyvalid,yetyieldinginconsistentresults.

wherered=longerthantriangleside,blue=shorter.

Selectionmethod1Chooseapointonthecircleandrotatethetrianglesothatthepointisatonevertex.Chooseanotherpointonthecircleanddrawthechordjoiningittothefirstpoint.Forpointsonthearcbetweentheendpointsofthesideoppositethefirstpoint,thechordislongerthanasideofthetriangle.Thelengthofthearcisonethirdofthecircumferenceofthecircle,thereforetheprobabilityarandomchordislongerthanasideoftheinscribedtriangleisonethird.

Selectionmethod2Choosearadiusofthecircleandrotatethetrianglesoasideisperpendiculartotheradius.Chooseapointontheradiusandconstructthechordwhosemidpointisthechosenpoint.Thechordislongerthanasideofthetriangleifthechosenpointisnearerthecenterofthecirclethanthepointwherethesideofthetriangleintersectstheradius.Sincethesideofthetrianglebisectstheradius,itisequallyprobablethatthechosenpointisnearerorfarther.Thereforetheprobabilityisonehalf.

Selectionmethod3Chooseapointanywherewithinthecircleandconstructachordwiththechosenpointasitsmidpoint.Thechordislongerthanasideoftheinscribedtriangleifthechosenpointfallswithinaconcentriccircleofradius1/2.Theareaofthesmallercircleisonefourththeareaofthelargercircle,thereforetheprobabilityisonefourth.

Bertrandintendedtoshowthattheclassicaldefinitionofprobabilityisnotapplicabletoaproblemwithaninfinityofpossibleoutcomes.Accordingtotheclassicaldefinition,theprobabilityofacompoundeventistheratioofthenumberoffavorablecasestothetotalnumberofcases.Suchadefinitio