计量经济学2ppt课件.ppt

CHAPTER2 AREVIEWOFBASICSTATISTICALCONCEPTS 2 1SOMENOTATION 1 TheSummationNotationcanbeabbreviatedas or2 PropertiesoftheSummationOperator 2 2EXPERIMENT SAMPLESPACE SAMPLEPOINT ANDEVENTS 1 ExperimentAstatistical randomexperiment aprocessleadingtoatleasttwopossibleoutcomeswithuncertaintyastowhichwilloccur 2 SamplespaceorpopulationThepopulationorsamplespace thesetofallpossibleoutcomesofanexperiment3 SamplePointSamplePoint eachmember oroutcome ofthesamplespace orpopulation 4 EventsAnevent acollectionofthepossibleoutcomesofanexperiment thatis itisasubsetofthesamplespace Mutuallyexclusiveevents theoccurrenceofoneeventpreventstheoccurrenceofanothereventatthesametime Equallylikelyevents oneeventisaslikelytooccurastheotherevent Collectivelyexhaustiveevents eventsthatexhaustallpossibleoutcomesofanexperiment 2 3RANDOMVARIABLES Arandom stochasticvariable r v forshort avariablewhose numerical valueisdeterminedbytheoutcomeofanexperiment 1 Adiscreterandomvariable anr v thattakesononlyafinite oraccountablyinfinite numberofvalues 2 Acontinuousrandomvariable anr v thatcantakeonanyvalueinsomeintervalofvalues 2 4PROBABILITY 1 TheClassicalorAPrioriDefinition Ifanexperimentcanresultinnmutuallyexclusiveandequallylikelyoutcomes andifmoftheseoutcomesarefavorabletoeventA thenP A theprobabilitythatAoccurs ism nTwofeaturesoftheprobability 1 Theoutcomesmustbemutuallyexclusive 2 Eachoutcomemusthaveanequalchanceofoccurring 2 RelativeFrequencyorEmpiricalDefinitionFrequencydistribution howanr v aredistributed Absolutefrequencies thenumberofoccurrenceofagivenevent Relativefrequencies theabsolutefrequenciesdividedbythetotalnumberofoccurrence EmpiricalDefinitionofProbability ifinntrials orobservations mofthemarefavorabletoeventA thenP A theprobabilityofeventA issimplytherationm n thatis relativefrequency providedn thenumberoftrials issufficientlylargeInthisdefinition wedonotneedtoinsistthattheoutcomebemutuallyexclusiveandequallylikely 2 4PROBABILITY 3 Propertiesofprobabilities 1 0 P A 1 2 IfA B C aremutuallyexclusiveevents then P A B C P A P B P C 3 IfA B C aremutuallyexclusiveandcollectivelyexhaustivesetofevents P A B C P A P B P C 1 2 4PROBABILITY Rulesofprobability 1 IfA B C areanyevents theyaresaidtobestatisticallyindependenteventsif P ABC P A P B P C P 2 IfeventsA B C arenotmutuallyexclusive thenP A B P A P B P AB ConditionalprobabilityofA givenBConditionalprobabilityofB givenA 2 5RANDOMVARIABLESANDPROBABILITYDISTRIBUTIONFUNCTION PDF Theprobabilitydistributionfunctionorprobabilitydensityfunction PDF ofarandomvariableX thevaluestakenbythatrandomvariableandtheirassociatedprobabilities 1 PDFofaDiscreteRandomVariableDiscreter v X takesonlyafinite orcountablyinfinite numberofvalues Probabilitydistributionorprobabilitydensityfunction PDF itshowshowtheprobabilitiesarespreadoverordistributedoverthevariousvaluesoftherandomvariableX PDFofadiscreter v X f X P X Xi fori 1 2 3 n0forX Xi 2 PDFofaContinuousRandomVariableTheprobabilityforacontinuousr v isalwaysmeasuredoveranintervalForacontinuousr v theprobabilitythatsuchanr v takesaparticularnumericalvalueisalwayszero 3 CumulativeDistributionFunction CDF F X P X x P X x theprobabilitythatther v Xtakesavalueoflessthanorequaltox wherexisgiven Infact aCDFismerelyan accumulation orsimplythesumofthePDFforthevaluesofXlessthanorequaltoagivenx CDFofadiscreter v stepfunction CDFofacontinuousr v continuouscurve 2 6MULTIVARIATEPROBABILITYDENSITYFUNCTIONS 1 Two variate BivariatePDFSingle UnivariateprobabilitydistributionfunctionsMultivariateprobabilitydistributionfunctionsTwo variate BivariatePDFJointprobability theprob thatther v XtakesagivenvalueandYtakesagivenvalue Bivariate jointPDF f X Y 1 DiscretejointPDF f X Y P X x Y y 0whenX x Y y 2 ContinuousjointPDF omitted 2 MarginalProbabilityDensityFunction TherelationshipbetweentheunivariatePDFs f X orf Y andthebivariatejointPDF ThemarginalprobabilityofX theprobabilitythatXassumesagivenvalueregardlessofthevaluestakenbyY ThemarginalPDFofX thedistributionofthemarginalprobabilities sumthejointprob correspondingtothegivenvalueofXregardlessofthevaluestakenbyY ThemarginalPDFofY sumthejointprob correspondingtothegivenvalueofYregardlessofthevaluestakenbyX 3 ConditionalProbabilityDensityFunction 1 discreteconditionalPDFSconditionalPDFofY f Y X P Y y X x conditionalPDFofX f X Y P X x Y y TheconditionalPDFofonevariable giventhevalueoftheothervariable issimplytheratioofthejointprobabilityofthetwovariablesdividedbythemarginalorunconditionalPDFoftheother i e theconditioning variable X of probability marginal the Y and X of probability joint the X f Y X f X Y F Y of probability marginal the Y and X of probability joint the Y f Y X f Y X F 2 6MULTIVARIATEPROBABILITYDENSITYFUNCTIONS 4 StatisticalIndependenceIndependentrandomvariables TwovariablesXandYarestatisticallyindependentifandonlyiftheirjointPDFcanbeexpressedastheproductoftheirindividual ormarginal PDFsforallcombinationsofXandYvalues f X Y f X f Y 2 7CHARACTERISTICS MOMENTSOFPROBABILITYDISTRIBUTIONS 1 ExpectedValue AMeasureofCentralTendency 1 DefinitionExpectedValue ameasureofcentraltendency givesthecenterofgravity Theexpectedvalueofadiscreter v isthesumofproductsofthevaluestakenbyther v andtheircorrespondingprobabilities 2 Propertiesofexpectedvalue Ifaandbareconstants then E b bE X Y E X E Y E XY E X E Y ifX Yareindependentr v E aX aE X E aX b aE X E b aE X b 2 7CHARACTERISTICS MOMENTSOFPROBABILITYDISTRIBUTIONS 2 Variance AMeasureofDispersion 1 DefinitionofthevarianceofX theexpectedvalueofthesquareddifferencebetweenanindividualXvalueanditsexpectedormeanvalue 2 ComputethevarianceDiscrete 3 PropertiesofvarianceIfaandbareconstantsvar a 0IfX Yaretwoindependentr v thenvar X Y var X var Y var X Y var X var Y var X b var X var aX a2var X var aX b a2var X var aX bY a2var X b2var Y 3 Covariance characteristicsofmultivariatePDFs 1 Definition Covarianceisameasureofhowtwovariablesvaryormovetogether Ifthetwor v moveinthesamedirection thenthecovcov X Y E X x Y y E XY x yDiscrete 2 PropertiesofcovarianceIfXandYareindependentr v thencov X Y 0Ifa b c dareconstants thencov a bX c dY bdcov X Y cov X X var X 4 CorrelationCoefficient 1 Definitionofthecorrelationcoefficient ameasureoflinearassociationbetweentwovariables i e howstronglythetwovariablesarelinearlyrelated 2 Propertiesofcorrelationcoefficient 1 1If0 1twovariablesarepositivelycorrelated 1 0negativelycorrelated 3 VariancesofcorrelatedvariablesIfX Yarenotindependent thatis theyarecorrelated thenvar X Y var X var Y 2cov X Y var X Y var X var Y 2cov X Y y x Y X s s r cov 5 ConditionalExpectationunconditionalexpectation conditionalexpectation6 TheSkewnessandtheKurtosisofaPDFmomentsofthePDFofar v Xfirstmoment E X xsecondmoment E X x 2thirdmoment E X x 3n thmo