数学专业英语12分析解析.docx
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数学专业英语12分析解析
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.Suchadefin