章节定量分析任务培训

QuantitativeAnalysis–QuantitativeAnalysis–Discreteandcontinuousprobabilitydistributions，BayesianEstimatingtheparametersofPopulationandsampleStatisticalinferenceandhypothesisLinearregressionwithsingleandmultipleTimeseriesCorrelationsandEstimatingcorrelationandvolatilityusingEWMAandGARCHSimulation2-ReadingsforQuantitativeReadingsforQuantitativeMichaelMiller,MathematicsandStatisticsforFinancialRiskManagement(Hoboken,NJ:JohnWiley&Sons,1933).•Chapter2-•Chapter3-BasicChapter4-•Chapter5-HypothesisTesting&ConfidenceJamesstockandMarkWatson,introductiontoeconometrics,Briefedition(Boston:PearsonEducation,2008).Chapter4-LinearregressionwithoneChapter5-Regressionwithasingleregressor:HypothesisTestsandconfidenceintervalsChapter6-LinearregressionwithmultipleChapter7-HypothesisTestsandconfidenceintervalsinmultiple3-ReadingsforQuantitative15.Johnhull,RiskReadingsforQuantitative15.Johnhull,RiskManagementandFinancialInstitutions,3rdEdition(Boston:PearsonPrenticehall,1932).Chapter11.Correlationsand16.Francisx.Diebold,ElementsofForecasting,4thEdition(Mason,Ohio:CengageLearning,2006).Chapter5.ModelingandForecastingTrend(Section5.4only—SelectingForecastingModelsUsingtheAkaikeandSchwarzChapter7.CharacterizingChapter8.ModelingCycles:MA,AR,andARMA17.JohnHull,Options,Futures,andOtherDerivatives,9thEditionork:PrenticeHall,1934).(New•Chapter23–EstimatingVolatilitiesandCorrelationsforRisk18.DessislavaPachamanovaandFrankFabozzi,SimulationandOptimizationinFinance(hoboken,NJ:JohnWiley&Sons,1930).Chapter4.Simulation4-如何学好定量分析•抓住基如何学好定量分析•抓住基本概•搞清来•忽略理论推•多做习5-FRMQuantitativeFRMQuantitativeProbabilityDistributions(Discrete&BasicHypothesisTestingandconfidenceLinearregressionwithoneRegressionwithasingleregressor:HypothesisTestsandconfidenceLinearregressionwithmultipleHypothesisTestsandconfidenceintervalsinmultipleElementsofCorrelationsandEstimatingVolatilitiesand Simulation 6-ProbabilityandProbabilityCertainProbabilityandProbabilityCertainUncertainRandomExperiment&RandomVariables:isanuncertainSampleSpaceorSampleRandomEvent:isasingleoutcomeorasetof•Mutuallyexclusiveevents:areeventsthatcannotbothhappenatthesameCollectivelyexhaustiveevents:arethosethatincludeallpossible•7-ProbabilityandProbabilityAims:describeandProbabilityandProbabilityAims:describeanddistinguishbetweencontinuousanddiscreterandomRandomRandomvariablesaredenotedbycapitallettersX,Y,Z,Thevaluestakenbythesevariablesareoftendenotedbysmallletters,x,y,z,etc.–outcomeDiscreterandomContinuousrandomProbabilityofanEvent:TheClassicalorAPriori8-P(A)numberofoutcomesfavorabletoAtotalnumberofoutcomesProbabilityandProbabilityVenn9- ProbabilityandProbabilityVenn9- AProbabilityandProbabilityAims:calculatetheprobabilityofaneventProbabilityandProbabilityAims:calculatetheprobabilityofaneventgivenadiscreteprobabilityfunction.Aims:distinguishbetweenindependentandmutuallyexclusiveevents.PropertiesofTheprobabilityofaneventalwaysliesbetween0and1.Thus,theprobabilityofeventA,P(A),satisfiesthisrelationship:IfA,B,C…aremutuallyexclusiveevents,theprobabilitythatanyoneofthemwilloccurisequaltothesumoftheprobabilitiesoftheirindividualIfA,B,C,…aremutuallyexclusiveandcollectivelyexhaustivesetofevents,thesumoftheprobabilitiesoftheirindividualoccurrencesis1.10-P(ABC...)P(A)P(B)P(C)...P(ABC...)P(A)P(B)P(C)0P(A)ProbabilityandProbabilityAims:definejointProbabilityandProbabilityAims:definejointprobability,describeaprobabilitymatrix,andcalculatejointprobabilitiesusingprobabilitymatrices.Aims:defineandcalculateaconditionalprobability,anddistinguishbetweenconditionalandunconditionalprobabilities.PropertiesofAdditionForeveryeventA,thereisaneventA’,calledthecomplementof11-P(AA')P(AA')P(AB)P(A)P(B)ProbabilityandProbabilityUnconditionalprobability:P(A),P(B)Conditionalprobability:P(A|B)ProbabilityandProbabilityUnconditionalprobability:P(A),P(B)Conditionalprobability:P(A|B)WewanttofindouttheprobabilitythattheeventAoccursknowingthattheeventBhasalreadyoccurred.ThisprobabilityiscalledtheconditionalprobabilityofAgivenB.TheconditionalprobabilityofA,givenB,isequaltotheratiooftheirjointprobabilitytothemarginalprobabilityofB.Inlikemanner,Jointprobability:P(AB)=P(A)P(B|A)=P(B)12-P(B|A)P(AB) P(A)P(A|B)P(AB) P(B)ProbabilityandProbabilityIndependentTheProbabilityandProbabilityIndependentTheoccurrenceofAhasnoinfluenceontheoccurrenceofBisindependentofP(AB)=P(A)P(B)P(B|A)=P(B)P(A|B)=ThreeeventsA1,A2,A3areindependentAk)P(Aj)P(Ak),jwherej,P(A1A2A3)P(1)P(2)P(A313-ProbabilityandProbabilityTotalProbabilityIfaneventAmustProbabilityandProbabilityTotalProbabilityIfaneventAmustresultinoneofthemutually,A1,A2,A3,...,AiAj=n∪ ii14- n P(A)(1)(AA1)P(A2)P(AA2)...P(An)P(AAnProbabilityandProbabilityX:Company’schoicedefault–{0,1}ProbabilityandProbabilityX:Company’schoicedefault–{0,1}NotAConditionalBNotCUnconditionalNot15-ProbabilityandProbabilityAims:describeProbabilityandProbabilityAims:describeBayes'sTheoremandapplythistheoreminthecalculationofconditionalprobabilities.Bayes’Prior16-P(A|B)P(B|A)P(ProbabilityandProbability快速诊断仪现一个人诊断为有病，问其真有病的概ProbabilityandProbability快速诊断仪现一个人诊断为有病，问其真有病的概有人没17-诊断没机器说机器说如果人如果人真没ProbabilityandProbabilityAims:defineanddistinguishbetweentheprobabilitydensityfunction,thecumulativedistributionfunction,andtheinversecumulativedistributionfunction,andcalculateprobabilitiesbasedoneachofthesefunctions.RandomProbabilityandProbabilityAims:defineanddistinguishbetweentheprobabilitydensityfunction,thecumulativedistributionfunction,andtheinversecumulativedistributionfunction,andcalculateprobabilitiesbasedoneachofthesefunctions.RandomVariablesandTheirProbabilityProbabilityDistributionofaDiscreteRandom•ProbabilityMassFunction(PMF)orProbabilityFunction•Propertiesofthen=3 012318-f(xi)x0f(xi)Binomial:n=3 f(Xxi)0,xf(Xxi)P(Xxi),i1,ProbabilityandProbabilityProbabilityDistributionofaProbabilityandProbabilityProbabilityDistributionofaContinuousRandomProbabilitydensityfunctionAPDFhasthefollowingThetotalareaunderthecurvef(x)isP(x1<X<x2)istheareaunderthecurvebetweenx1andx2.P(x1Xx2)P(x1Xx2)Xx2)Xx219-P(x1Xx2)fProbabilityandProbabilityCumulativeDistributionFunction1}0abxa0b20-P(aProbabilityandProbabilityCumulativeDistributionFunction1}0abxa0b20-P(aXb)=F(b)-xf(x)=Xb)=Areaundertweenaandbb)-F(X)P(XProbabilityandProbabilityPropertiesofProbabilityandProbabilityPropertiesof•F(-∞)=0andF(∞)=1,whereF(-∞)andF(∞)arethelimitsofF(X)asxtendsto-∞and∞,respectively.F(X)isanon-decreasingfunctionsuchthatifx2>x1thenP(X≥k)=1-F(k);thatis,theprobabilitythatXassumesavalueequaltoorgreaterthankis1minustheprobabilitythatXtakesavaluebelowk.P(x1≤X≤x2)=F(x2)-•••21-ProbabilityandProbabilityMultivariateprobabilitydensityWetakeXfrom1or2withthesameProbabilityandProbabilityMultivariateprobabilitydensityWetakeXfrom1or2withthesameprobability.WetakeYfrom[1,X]withthesameprobability.XY1212Definition:f(X,Y)=P(X=xandPropertiesofthebivariateorjointprobabilitymassfunctionf(X,Y)≧0forallpairsofXandY.Thisisbecauseallprobabilitiesarenonnegative.∑∑f(X,Y)22-ProbabilityandProbabilityMarginalProbabilityyfProbabilityandProbabilityMarginalProbabilityyf(Y)f(X,Yforallx23-MarginalprobabilitydistributionofXandYValueofXf(X)ValueofY itionofMarginalprobabilityf(X)f(X,Y)forallProbabilityandProbabilityStatisticalIndependence:f(X,Y)=StatisticalIndependenceoftworandom 12ProbabilityandProbabilityStatisticalIndependence:f(X,Y)=StatisticalIndependenceoftworandom 1231123YEXAMPLE:FRMEXAM2007–ThejointprobabilitydistributionofrandomvariablesXandYisgivenbyf(x,y)=kxyforx=1,2,3,y=1,2,3andkisapositiveconstant,whatistheprobabilitythatX+Ywillexceed5?A.B.C.D.Cannotbe24-CharacteristicsofProbabilityExpectedValue:AMeasureCharacteristicsofProbabilityExpectedValue:AMeasureofCentralTendency–thefirstPropertiesofExpected•••1.Ifbisaconstant,E(b)=b2.E(X+Y)=E(X)+E(Y)3.Ingeneral,E(XY)≠E(X)E(Y);IfXandYareindependentrandomvariables,thenE(XY)=E(X)E(Y)4.E(X2)Ifaisaconstant,Ifaandbareconstants,then•••25-E(X)xfE(X)xf(x)x1P(x1)x2P(x2) xnP(xnXCharacteristicsofProbabilityExpectedValueCharacteristicsofProbabilityExpectedValueofMultivariateProbabilityIntheabivariatePMF,itcanbeshownContinuingwithourX,Yexample,andapplyingaboveE(XY)(110.50)(120.00)(210.25)(2226-E(XY)xyf(XY CharacteristicsofProbabilityVariance:aMeasureofCharacteristicsofProbabilityVariance:aMeasureofDispersion–thesecondThedefinitionofThepositivesquarerootofVAR(X),x,isknownasthestandardTocomputethevariance,weusethefollowing27-VAR(X)E(X2)[E(XVAR(X)(xx)2fVAR(X)(X)2P(X x E(X )2XCharacteristicsofProbabilityCharacteristicsofProbability28-Giventhefollowingdata,whatisthebestestimateofitsexpectationanditsvariance? CharacteristicsofProbabilityPropertiesof•ThevarianceofCharacteristicsofProbabilityPropertiesof•Thevarianceofaconstantiszero.Bydefinition,aconstanthasnoIfXandYaretwoindependentrandomvariables,•VAR(X+Y)=VAR(X)+VARVAR(X-Y)=VAR(X)+VAR••••Ifbisaconstant,then:VAR(X+b)=VARIfaisconstant,then:VAR(aX)=a2VARIfaandbareconstant,then:VAR(aX+b)=a2VARIfXandYareindependentrandomvariablesandaandbareconstants,thenVAR(aX+bY)=a2VAR(X)+b2VAR(Y)Forcomputationalconvenience,wecanget:VAR(X)=E(X2)-[E(X)]2•E(X2)x2f(Xx29-CharacteristicsofProbabilityCoefficientofCharacteristicsofProbabilityCoefficientofExample:CoefficientofYouhavejustbeenpresentedwithareportindicatesthatthemeanmonthlyreturnonT-billsis0.25%withastandarddeviationof0.36%,andthemeanmonthlyreturnfortheS&P500is1.09%withastandarddeviationof7.03%.YourunitmanagerhasaskedyoutocomputetheCVforthesetwoinvestmentsandtointerpretyour30-CVxCharacteristicsofProbability•CovariancemeasureshowCharacteristicsofProbability•Covariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.Covariancerangesfromnegativeinfinitytopositive•Propertiesof•••••IfXandYareindependentrandomvariables,theircovarianceiscov(a+bX,c+dY)bdcov(X,YIfXandYareNOTindependent,var(XY)var(X)var(Y)2cov(X,Y31-cov(X,Y)E[(X-E(X))(Y-E(Y))]E(XY)-CharacteristicsofProbabilityEXAMPLE1:FRMCharacteristicsofProbabilityEXAMPLE1:FRMEXAM2007–SupposethatAandBarerandomvariables,eachfollowsastandardnormaldistribution,andthecovariancebetweenAandBis0.35,whatisthevarianceof3A+2B?B.C.D.EXAMPLE2:FRMEXAM2002–Giventhatxandyarerandomvariables,anda,b,canddareconstant,whichofthefollowingdefinitioniswrong?E(ax+by+c)=aE(x)+bE(y)+c,ifxandyareV(ax+by+c)=V(ax+by)+c,ifxandyarecov(ax+by,cx+dy)=acV(x)+bdV(y)+(ad+bc)cov(x,y),ifxandyareV(x-y)=V(x)+V(y)=V(x+y),ifxandyare••••32-CharacteristicsofProbabilityCorrelationPropertiesofCorrelationCharacteristicsofProbabilityCorrelationPropertiesofCorrelation•CorrelationmeasuresthelinearrelationshipbetweentworandomCorrelationhasnounits,rangesfrom–1toIftwovariablesareindependent,theircovarianceiszero,therefore,thecorrelationcoefficientwillbezero.Theconverse,however,isNOTtrue.Forexample,Y=X2Variancesofcorrelated•••33-var(XY)var(X)var(Y)2x InterpretationsofCorrelation34-Correlationr=perfectInterpretationsofCorrelation34-Correlationr=perfectpositive0<r<positivelinearr=nolinear−1<r<negativelinearr=perfectnegativeInterpretationsofCorrelation•InterpretationsofCorrelation••••EXAMPLE1:WhichofthefollowingstatementsaboutcorrelationcoefficientisItalwaysrangesfrom-1toAcorrelationcoefficientofzeromeansthattworandomvariablesareItisameasureoflinearrelationshipbetweentworandomItcanbecalculatedbyscalingthecovariancebetweentworandom•••••EXAMPLE2:IfXandYareindependentrandomthecovariancebetweenthetwovariablesisequaltothecorrelationbetweenthetwovariablesisequalto-thecovarianceandcorrelationbetweenthetwovariablesarebothequaltothevariablesareperfectlyfortwo(possiblydependent)randomvariables,XandY,anupperboundonthecovarianceofXandYis135-CharacteristicsofProbabilityChebyshev’s之内的概率不小于1-1/k²，对任意k>1。P(|XCharacteristicsofProbabilityChebyshev’s之内的概率不小于1-1/k²，对任意k>1。P(|X|k)11/k2,kofthemean36-23411113 11118 111115 CharacteristicsofProbabilitySkewness-AmeasureofasymmetryofaPDF-the3rdCharacteristicsofProbabilitySkewness-AmeasureofasymmetryofaPDF-the3rdNegative-Positive-==37-EX thirdmomentaboutS cubeofstandardxCharacteristicsofProbabilityKurtosis-AmeasureoftallnessorflatnessofaPDF-the4thKCharacteristicsofProbabilityKurtosis-AmeasureoftallnessorflatnessofaPDF-the4thKE(XfourthmomentaboutfourthpowerofstandardForanormaldistributiontheKvalueis3,andsuchaPDFisExcesskurtosis=kurtosis-Leptokurticvs.38-ExcesssameFatThinCoskewnessandThethirdcrosscentralCoskewnessandThethirdcrosscentralmomentisknownascoskewnessandthefourthcrosscentralmomentisknownascokurtosis.Assumefourseriesoffundreturns(ABCD)wherethemean,standarddeviation,skew,kurtosisallthesame,butonlytheorderofreturnsisThetwoportfolios(A+BandC+D)thesamemeananddeviation,buttheskewsoftheportfoliosare39-CoskewnessandScatterplotsshowthedifferenceCoskewnessandScatterplotsshowthedifferencebetween(BversusA)and(Dversus•AandB:theirbestpositivereturnsoccurduringthesametimeperiod,buttheirworstnegativereturnsoccurindifferentperiods.Thiscausesthedistributionofpointstobeskewedtowardthetop-rightoftheCandD:theirworstnegativereturnsoccurinthesameperiod,buttheirbestpositivereturnsoccurindifferentperiods.Inthesecondchart,the•pointsareskewedtowardthebottom-leftthe40-CoskewnessandThereasontheabovechartslookdifferentorthereasonthereturnsthetwoportfoliosCoskewnessandThereasontheabovechartslookdifferentorthereasonthereturnsthetwoportfoliosaredifferent,isbecausethecoskewnessbetweentheportfoliosisdifferent.••TheCoskewnessoftwovariables:TheCokurtosisoftwovariables:and、KAABBand41-AandCand--FRMQuantitativeFRMQuantitativeProbabilityDistributions(Discrete&BasicHypothesisTestingandconfidenceLinearregressionwithoneRegressionwithasingleregressor:HypothesisTestsandconfidenceLinearregressionwithmultipleHypothesisTestsandconfidenceintervalsinmultipleElementsofCorrelationsandEstimatingVolatilitiesand Simulation 42-TheAimsofTheAimsofthisAims:defineanddistinguishbetweenparametricandnonparametricAims:describekeypropertiesoftheuniformdistribution,Bernoullidistribution,Binomialdistribution,Poissondistribution,normaldistribution,andlognormaldistribution,andidentifycommonoccurrencesofeachdistribution.Aims:describeandapplythecentrallimittheoremandexplaintheconditionsforthetheorem.Aims:describethepropertiesofindependentandidenticallydistributed(i.i.d)Aims:describethepropertiesoflinearcombinationsofnormallydistributedAims:identifythekeypropertiesandparametersoftheChi-squared,Student'st,andF-distributions.43-SomeImportantProbabilityBernoullirandomP(Y=0)=1-BinomialrandomtheprobabilitySomeImportantProbabilityBernoullirandomP(Y=0)=1-BinomialrandomtheprobabilityofxsuccessesinnExpectationsand44-Bernoullirandomvariablepp(1-Binomialrandomvariablenp(1-p(x)P(Xx)npx(1 xSomeImportantProbabilityTheCumulativeBinomialProbabilityxSomeImportantProbabilityTheCumulativeBinomialProbabilityx01234545-F(x)P(Xx)alliP(X)=F(x)-F(xDerivingIndividual ForfromCumulativeP(3)F(3)F.813SomeImportantProbabilityTheBinomialDistribution-p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandas46-xxSomeImportantProbabilityTheBinomialDistribution-p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandas46-xxxBinomialProbability:n=10xxxxxSomeImportantProbabilityPoissonWhenthereareSomeImportantProbabilityPoissonWhentherearealargenumberoftrialsbutasmallprobabilitysuccess,weoftenusethePoisson••k:thenumberofsuccessper:theaverageofexpectednumberofsuccessper••••thenumberoffishcaughtinathenumberofpotholesona1kmstretchofthenumberofpersonsappearedinashoppingmallthenumberofphonecallsinaday47- p(k)P(Xk) ,kSomeImportantProbabilitySomeImportantProbabilityAcompanyreceivesthreecomplaintsperdayonaverage.Whatistheprobabilityofreceivingmorethanonecomplaintonaparticularday?=“morethanone”meansthatk=2or3or4orP(‘morethanone’)=P(2)+P(3)+P(4)+P(‘morethanone’)=1–{P(0)+P(1)}P(0)=e-3×30/0!=0.0498P(1)=e-3×31/1!=P(0)+P(1)=P(‘morethanone’)=1–{P(0)+P(1)}=1–0.1992=48-SomeImportantProbabilityPropertiesofPoissonSomeImportantProbabilityPropertiesofPoissonE(X)=D(X)=ThesumofindependentPoissonvariablesisafurtherPoissonvariablewithmeanequaltothesumoftheindividualmeans.ThePoissondistributionprovidesanapproximationfortheBinomialdistribution,whennislargeandpissmall,=np•••When•thenthePoissondistributionconvergesthenormal49-SomeImportantProbabilityContinuousUniformProbabilitydensity1,axSomeImportantProbabilityContinuousUniformProbabilitydensity1,axf(x)b,abCumulativedistributionforxforaxforx0F(x)xab50-SomeImportantProbabilityPropertiesSomeImportantProbabilityPropertiesofContinuousUniformE(X)=(a+Var(X)=(b–Foralla≤x1<x2≤b,wexP(x1Xx2)f(x)dx(x2x1)/(bTherandomvariableXwithdensityfunctionf(x)=k/3for2<=x<=8,and0otherwise.Calculateitsmean.51-SomeImportantProbabilityNormalAsnincreases,thebinomialdistributionapproachesNormaln=n=n=NormalProbabilityDensity211xf(x)exp2...and3 e252-SomeImportantProbabilityNormalAsnincreases,thebinomialdistributionapproachesNormaln=n=n=NormalProbabilityDensity211xf(x)exp2...and3 e252- xBinomialDistribution:n=14,012345678910111213xBinomialDistribution:n=10,0123456789xBinomialDistribution:n=6, xSomeImportantProbabilityTheShapeoftheNormalSomeImportantProbabilityTheShapeoftheNormalThenormalcurveThetwohalvesTheoretically,thecurveextendsto-Themean,median,andmodeareX~N(,2•••,Fullydescribedbyitstwoµand2Bell-shaped,Symmetricaldistribution:skewness=0;Alinearcombinationoftwo(ormore)independentnormallydistributionrandomvariablesisnormallydistributed.53-ibutionDensityTheoretically,thecurveextendstoSomeImportantProbabilityTheSomeImportantProbabilityTheconfidence••••Approximately68%ofallobservationsfallintheintervalApproximately90%ofallobservationsfallintheintervalμ±1.65σApproximately95%ofallobservationsfallintheintervalApproximately99%ofallobservationsfallintheinterval54-SomeImportantProbabilityThestandardnormal•••N(0,1)orStandardization:SomeImportantProbabilityThestandardnormal•••N(0,1)orStandardization:ifX~N(µ,σ²),thenZX~HowtouseZ-Howweusethestandardnormaldistributiontocompute••••(79)FirstZ75.9701.96,thencomputeP(Z1.96)10.9753Question1:computetheprobability64.12XQuestion2:computetheprobabilityof64.12XandX55-NormalEXAMPLE1:FRMEXAM2005–LetZbeastandardnormalrandomvariable,andeventXisdefinedtohappenifeitherZtakesavaluebetweenNormalEXAMPLE1:FRMEXAM2005–LetZbeastandardnormalrandomvariable,andeventXisdefinedtohappenifeitherZtakesavaluebetween-0.5and+0.5orZtakesanyvaluegreaterthen1.5.WhatistheprobabilityofeventXhappeningifN(0.5)=0.6915andN(-1.5)=0.0668,whereN(.)isthecumulativedistributionfunctionofastandardnormal••••A.B.C.D.EXAMPLE2:FRMEXAM2003–WhichofthefollowingstatementaboutthenormaldistributionisnotKurtosisequalsSkewnessequalsTheentiredistributioncanbecharacterizedbytwomoments,meanand1exp1x2•D.Thenormaldensityfunctionhasthefollowingexpression:.f(x)56-NormalEXAMPLE3:FRMEXAM2006–NormalEXAMPLE3:FRMEXAM2006–Whichtypeofdistributionproducesthelowestprobabilityforavariabletoexceedaspecialextremevaluewhichisgreaterthanthemean,assumingthedistributionallhavethesamemeanand••••Aleptokurticdistributionwithakurtosisof4.Aleptokurticdistributionwithakurtosisof8.Anormaldistribution.AplatykurticdistributionEXAMPLE4:A$50millionprudentfund(PF)ismergedwitha$200millionaggressivefund(AF).ThereturnofPF~N(0.03,0.072)andthereturnofAF~N(0.07,0.152).Seniormanageraskedyoutoestimatethelikelihoodthatthereturnsofthecombinedportfoliowillexceed26%.Assumingthereturnsareindependent,whatistheprobabilitythatthereturnwillexceed•A.B.C.D.57-TheBlack-ScholesModelassumesthatthepriceoftheunderlyingassetislognormallydistributedIflnXisnormal,thenTheBlack-ScholesModelassumesthatthepriceoftheunderlyingassetislognormallydistributedIflnXisnormal,thenXisRightBoundedfrombelowbylnX~N(,21EX 22VXexp222exp22112lnx2,xf(x) 58-NormalEXAMPLE1:FRMNormalEXAMPLE1:FRMEXAM1999–Whichofthefollowingstatementsbestcharacterizestherelationshipbetweenthenormalandlognormaldistributions?lognormaldistributionsisthelogarithmofthenormalIFthelogoftherandomvariableXislognormallydistributed,thenXisnormallydistributed.IfXislognormallydistributed,thenthenaturallogofXisnormallyThetwodistributionshasnothingtodowithoneanother.EXAMPLE2:FRMEXAM2007–Q21Theskewofalognormaldistributionis03••••••••59-SomeImportantProbabilityRandomSamplingfromaNormal•RandomnumberXThesamplingorprobabilitydistributionofthe••RandomsamplingSomeImportantProbabilityRandomSamplingfromaNormal•RandomnumberXThesamplingorprobabilitydistributionofthe••Randomsampling:i.i.d(independentlyandidenticallySamplingdistributionofan•,isarandomsamplefromanormalpopulationIfX1,X2,thenthesample Xalsofollowsmeanand2Xnormaldistributionwiththesamebutwith XnthatX~N(X Xn60-Z(XX)~N(0,X SomeImportantProbabilityThecentrallimittheoremLaplace:ifX1,X2,isarandomsamplefromanypopulation,probabilitydistribution)withSomeImportantProbabilityThecentrallimittheoremLaplace:ifX1,X2,isarandomsamplefromanypopulation,probabilitydistribution)withmean2X,thesampleX2Xtendstobenormallydistributedwithmean andasnsamplesizeincreasesindefinitely(technically,infinitely30X~NnStandardError(se)ofmeanXnHowever,thepopulation’sstandarddeviationisalmostneverInstead,weusethestandarddeviationofthesample XXSxin61-NormalAreturnserieswith250NormalAreturnserieswith250observationshasasamplemeanof10%andastandarddeviationof15%.ThestandarderrorofthesamplemeanisclosettoA.B.C.D.TheweightofpineappleconformsanormaldistributionofN(0.5kg,(0.1kg)2).Thepineappleisindependentofeachother.Whatistheprobabilitythattheweightof5000pineappleswillexceed2510kg?A.B.C.D.••••62-SomeImportantProbabilityTheChi-Square(2SomeImportantProbabilityTheChi-Square(2)ProbabilitynX~N(0,1), ~22ii63-SomeImportantProbabilityPropertiesoftheChi-Square•SomeImportantProbabilityPropertiesoftheChi-Square•Thechi-squaredistributiontakeonlypositivevalueandrangesfrom0toinfinity(afterall,itisthedistributionofasquaredquantity).Forcomparativelyfewd.f.thedistributionishighlyskewedtotheright,butasthed.f.increase,thedistributionbecomesincreasinglysymmetricalandapproachesthenormaldistribution.E(X)=k,D(X)=2k,wherekisthed.f.Thisisanote-worthypropertyofthechi-squaredistributioninthatitsvarianceistwiceitsmeanvalue.IfZ1andZ2aretwoindependentchi-squarevariableswithk1andk2thentheirsum(Z1+Z2)isalsoachi-squarevariablewithd.f.=(k1+k2•••(n~2(n2064-SomeImportantProbabilityThetDistribution(Student’sZ(XX)~Nboth andSomeImportantProbabilityThetDistribution(Student’sZ(XX)~Nboth and••Recall1)are2xX Supposeweonlyknow2xandbyits(sample)(X-Xi2,weobtainanewS2xn65-X~N(0,1);Y~2且X,Y独立，t ~Yt=X ~S xSomeImportantProbabilitySomeImportantProbabilityPropertiesofthet••••Themeanoftdistributioniszero,anditsvariancen/(n-2)Lesspeakedthananormaldistribution(“fattertails”)Asthedegreesoffreedomgetslarger,theshapeoft-approachesstandardnormal66-SomeImportantProbabilityTheFDistribution(VarianceRatioU~SomeImportantProbabilityTheFDistribution(VarianceRatioU~(n),V~2(n且U,V独212FU/~F(n,n V/267-FS ~(m1,S SomeImportantProbabilityPropertiesofFSkewedtotherightandalsoSomeImportantProbabilityPropertiesofFSkewedtotherightandalsorangesbetween0andApproachesthenormaldistributionbecomeand1FF~F(n,n)~F(n,n 1F(n,n F(n,n 68-FRMQuantitativeFRMQuantitativeProbabilityDistributions(Discrete&BasicHypothesisTestingandconfidenceLinearregressionwithoneRegressionwithasingleregressor:HypothesisTestsandconfidenceLinearregressionwithmultipleHypothesisTestsandconfidenceintervalsinmultipleElementsofCorrelationsandEstimatingVolatilitiesand Simulation 69-TheAimsofthisTheAimsofthisAims:define,calculateandinterpretthemean,standarddeviationandvarianceofarandomvariable.Aims:definecalculateandinterpretthecovarianceandcorrelationbetweentworandomvariables.Aims:interpretandcalculatethevarianceofaportfolio,andunderstandthedeviationoftheminimumvariancehedgeratio.Aims:calculatethemeanandvarianceofsumsofthelargervariables.Aims:describefourcentralmomentsofstatisticalvariableordistribution:mean,varianceskewnessandkurtosis.Aims:interprettheskewnessandkurtosisofastatisticaldistribution,andinterprettheconceptsofcoskewnessandcokurtosis.Aims:defineandinterpretthebestlinearunbiased70-StatisticalInference:EstimationHypothesisWhatisStatisticalStatisticalInference:EstimationHypothesisWhatisStatistical•Concernedwithdrawingconclusionsaboutthenatureorsomepopulation(e.g.,thenormal)onthebasisofarandomsamplethatsupposedlybeendrawnfromthatpopulation.Looselyspeaking,isthestudyoftherelationshipbetweenapopulationandasampledrawnfromthatpopulation.•Samplingandpopulationsamplestatistic71-CharacteristicsofProbabilityFromthePopulationtotheCharacteristicsofProbabilityFromthePopulationtotheSample•ThesamplemeanofarandomvariableXisgenerallydenotedbythesymbol(readasXbar)andisdefined•ThesamplemeanisknownasanestimatorofE(X),whichwecannowcallthepopulationmean.Anestimateofthepopulationissimplythenumericalvaluetakenbyanestimator.•72- X X CharacteristicsofProbabilitySampleThesamplevariance,denotedbyCharacteristicsofProbabilitySampleThesamplevariance,denotedbyS2whichisanestimator,2xxwhichwecannowcallthepopulationvariance.ThesamplevarianceisdefinedasIfthesamplesizeisreasonablylarge,wecandividebyninsteadofTheexpression(n-1)isknownasthedegreesofiscalledthesamplestandard73- nS2 (XX n1 StatisticalInference:EstimationandHypothesisEstimationandHypothesisTesting:TwinBranchesStatisticalInference:EstimationandHypothesisEstimationandHypothesisTesting:TwinBranchesOfStatisticalPricetoearning(P/E)ratiosof28companiesontheNewYorkstockexchange(NYSE).standardSource:www.Stockselector.74-StatisticalInference:EstimationandHypothesisEstimationof1.PointStatisticalInference:EstimationandHypothesisEstimationof1.PointUsingasinglenumericalvaluetoestimatetheparameterofX2XX2.ConfidenceintervalUsinganintervaltoestimatethescopeofthe75-StatisticalInference:EstimationandHypothesisPropertiesofpointUnbiased-ThemeanoftheestimatorscoincideswiththetrueStatisticalInference:EstimationandHypothesisPropertiesofpointUnbiased-ThemeanoftheestimatorscoincideswiththetrueparameterE(X)Anunbiasedestimatorisalsoefficientifthevarianceofitssam