一级基础定量分析

LinearLinearRegressionwithOneLinearRegressionwithMultipleSimulationEstimatingVolatilitiesand3–1.BasisRandomOutcome&ImpossibleeventBasisRandomOutcome&Impossibleevent、CertaineventseventsPopulation&RandomDiscreterandomvariables、ContinuousrandomProbability&Probabilityalgorithm(概率运算法则ConceptofMultiplicationrule、AdditionTotalProbabilityFormula、Bayes’ProbabilityFunction&CumulativeDistributionDiscreterandomvariables、Continuousrandom4–Randomexperiment(随机试验AnobservationormeasurementRandomexperiment(随机试验AnobservationormeasurementOutcome(结果Theresultofasingletrial.processwithmultiplebutexample,ifwerolltwodices,outcomemightbe3and4;adifferentoutcomemightbe5andEvent(事件Theresultthatreflectsnone,one,ormoreoutcomesinthesamplespace.Eventscanbesimpleorcompound.Aneventisasubsetofthesamplespace.Ifwerolltwodices,anexampleofaneventmightberolling7intotal.Mutuallyexclusiveevents(互斥事件):EventsthatcannothappenatthesameExhaustiveevents(完备事件):5–ProbabilityofanTwodefiningpropertiesof0ProbabilityofanTwodefiningpropertiesof0≤P(E)≤IfE1,E2,……,Enismutuallyexclusiveandexhaustive,then:P(E1)+P(E2)+……+P(En)=1Venn6– APAnumberofoutcomesfavorabletoAtotalnumberofoutcomesJointTheprobabilitythattherandomvariables(inJointTheprobabilitythattherandomvariables(inthiscase,bothrandomvariables)takeoncertainvaluessimultaneously,P(AB).UnconditionalProbability(边际概率,a.k.amarginalTheexpectedvalueofthevariablewithoutanyrestrictions(orlackinganypriorinformation),P(A).ConditionalProbability(条件概率Anexpectedvalueforthevariableconditionalonpriorinformationa conditionalexpectationofB,conditionalonA,isgivenby7–ProbabilityandProbabilityUnconditionalProbabilityandProbabilityUnconditionalprobability:P(A),Conditionalprobability:8–P(B|A)P(AB);P(A)P(A|B)P(AB);P(B)0ProbabilityandProbabilityJointprobability:MultiplicationProbabilityandProbabilityJointprobability:MultiplicationP(AB)=P(A|B)×P(B)=P(B|A)×IfAandBaremutuallyexclusiveevents,P(AB)=P(A|B)=P(B|A)=ProbabilitythatatleastoneoftwoeventswillAdditionP(AorB)=P(A)+P(B)–IfAandBaremutuallyexclusiveevents,P(AorB)=P(A)+9–ProbabilityandProbabilityProbabilityandProbabilityTheoccurrenceofAhasnoinfluenceofontheoccurrenceofP(A|B)=P(A)orP(B|A)=P(AB)=P(A)×P(AorB)=P(A)+P(B)–IndependenceandMutuallyExclusivearequiteIfexclusive,mustnotCauseexclusivemeansifAoccur,Bcannotoccur,AinfluentsP(AB)=P(A)×10–ProbabilityandProbabilityTotalProbabilityIfaneventProbabilityandProbabilityTotalProbabilityIfaneventAmustresultinoneofthemutuallyexclusiveeventsA1,A2,A3,……,An,then(1)AA=(i≠ n∪Ai11– PAPA1PAA1PA2PAA2...PAnPAAnProbabilityandProbabilityX:Company’schoicedefault–{0,1}ANotProbabilityandProbabilityX:Company’schoicedefault–{0,1}ANotBConditionalNotCUnconditionalNot12–ProbabilityandProbabilityBayes’Prior一个人有病的概率是10%，没病的概率是90%。在有ProbabilityandProbabilityBayes’Prior一个人有病的概率是10%，没病的概率是90%。在有病的情况下机器诊断出有病的概率是99%，诊断出没病的概率是1%；在没病的情况下机器诊断出有病的概率是5%，诊断出没病的概率是95%。若机器诊断出有病的情况下人真的有病的概率是多少13–机器说机器说如果人如果人PB|PA|B PAPBProbabilityandProbability人ProbabilityandProbability人14–P(A|B) 99%10% 68.75%99%10%5%90%ProbabilityandProbabilityProbabilityDescribetheprobabilitiesofProbabilityandProbabilityProbabilityDescribetheprobabilitiesofallthepossibleoutcomesDiscreteandcontinuousrandomforaDiscreterandomvariables:thenumberofpossibleoutcomescanbecounted,andforeachpossibleoutcome,thereisameasurableandpositiveContinuousvariables:thenumbereveniflowerandupperboundsP(x)=0eventhoughxcanP(x1<X<15–ProbabilityandProbabilityProbabilityfunction:p(x)=ProbabilityandProbabilityProbabilityfunction:p(x)=P(X=Fordiscreterandom0≤p(x)≤Σp(x)=Probabilitydensityfunction(p.d.f):ForcontinuousrandomvariableCumulativeprobabilityfunction(c.p.f):16–xF(x)F(x)=P(X≤ProbabilityandProbabilityRandomVariablesandTheirProbabilityProbabilityDistributionofaDiscreteRandomProbabilityMassFunctionProbabilityandProbabilityRandomVariablesandTheirProbabilityProbabilityDistributionofaDiscreteRandomProbabilityMassFunction(PMF)orProbabilityFunctionPropertiesoftheForn=3p=Binomial:n=3,p= 0123 X0317–f(xi)x0f(xi)f(Xxi)0,xf(Xxi)P(Xxi),i,,3...ProbabilityandProbabilityProbabilityDistributionofaContinuousRandomProbabilitydensityfunctionProbabilityandProbabilityProbabilityDistributionofaContinuousRandomProbabilitydensityfunctionProbabilitythatheightliesbetween60and68X0HeightinAPDFhasthefollowingThetotalareaunderthecurvef(x)isP(x1<X<x2)istheareaunderthecurvebetweenx1andP(x1Xx2)P(x1Xx2)P(x1Xx2)P(x1Xx218–ProbabilityP(x1Xx2)ProbabilityandProbabilityCumulativeDistributionFunction10abx0abProbabilityandProbabilityCumulativeDistributionFunction10abx0abx19–P(a≤X≤b)=Areaunderf(x)betweenaandb=F(b)–P(a≤X≤b)=F(b)–F(X)P(XProbabilityandProbabilityProbabilityandProbabilityPropertiesofF(-∞)=0andF(+∞)=F(X)isanon-decreasingfunctionsuchthatifP(X≥k)=1–P(x1≤X≤x2)=F(x2)–>thenF(x2)≥20–ProbabilityandProbabilityMultivariateprobabilitydensityWetakeXfromProbabilityandProbabilityMultivariateprobabilitydensityWetakeXfrom1or2withthesameprobability.WetakeYfrom[1,X]withthesameprobability.XY1212Definition:f(X,Y)=P(X=xandPropertiesofthebivariateorjointprobabilitymassfunctionf(X,Y)≥0forallpairsofXandY.Thisisbecauseallprobabilitiesare∑∑f(X,Y)21–ProbabilityandProbabilityMarginalprobabilitydistributionofXandValueofProbabilityandProbabilityMarginalprobabilitydistributionofXandValueofValueof1122Definitionofmarginalprobability22–f(Y)f(X,Y)forallxf(X)f(X,Y)forallyProbabilityandProbabilityStatisticalDefinitionofStatisticalIndependence:f(X,Y)=23–XY1ProbabilityandProbabilityStatisticalDefinitionofStatisticalIndependence:f(X,Y)=23–XY1231231ExampleThejointprobabilitydistributionofrandomvariablesXandYisgivenbyf(x,y)=ExampleThejointprobabilitydistributionofrandomvariablesXandYisgivenbyf(x,y)=kxyforx=1,2,3,y=1,2,3andkisapositiveconstant,whatistheprobabilitythatX+YwillexceedCannotbe24–ExampleaprincipalbalanceExampleaprincipalbalanceofhalfofthesubprimemortgagesandone-quarterofthenon-subprimemortgagesexceedsthevalueofthepropertyusedascollateral.Ifyourandomlyselectamortgagefromtheportfolioforreviewanditsprinciplebalanceexceedsthevalueofthecollateral,whatistheprobabilitythatitisasubprime25–HypothesisTestsandConfidenceHypothesisTestsandConfidenceIntervalsLinearRegressionLinearRegressionwithOneLinearRegressionwithMultipleSimulationEstimatingVolatilitiesand26–2.BasicExpectedExpectedAmeasureofcentralExpectedExpectedAmeasureofcentraltendency–thefirstPropertiesofExpectedIfbisaconstant,E(b)=Ifaisaconstant,E(aX)=Ifaandbareconstants,thenE(aX+b)=aE(X)+E(b)=aE(X)+b.E(X2)≠[E(X)]2E(X+Y)=E(X)+Ingeneral,E(XY)≠E(X)E(Y);IfXandYareindependentrandomvariables,thenE(XY)=E(X)E(Y).27–E(X)E(X)P(xi)xiP(x1)x1P(x2)x2...P(xnAmeasureofdispersion–thesecondAmeasureofdispersion–thesecondAboveformulaisthedefinitionofvariance.TocomputetheweusethefollowingMeasureshownoisyorunpredictablethatrandomvariableThepositivesquarerootofσ2,σ,isknownasthestandardxxalsocalled28–σ2EX2EX2EX22EXPropertiesofThevarianceofaconstantiszero.Bydefinition,aconstantIfaisconstant,then:σPropertiesofThevarianceofaconstantiszero.Bydefinition,aconstantIfaisconstant,then:σ2(aX)=Ifbisaconstant,then:σ2(X+b)=Ifaandbareconstant,then:σ2(aX+b)=IfXandYaretwoindependentrandomvariables,IfXandYareindependentrandomvariablesconstants,thenσ2(aX+bY)=a2σ2(X)+aandbForcomputationalconvenience,wecanget:σ2(X)=E(X2)–EX2x2px. 29–σ2(X–Y)=σ2(X)+σ2(X+Y)=σ2(X)+SampleSampleThesamplemeanSampleSampleThesamplemeanofar.v.XisdefinedXThesamplemeanisknownasannowcallthepopulationofE(X),whichweAnestimateofthepopulationissimplythenumericalvaluetakenan30–i1SampleSampleThesamplevariance,denotedwhichisSampleSampleThesamplevariance,denotedwhichisanestimator,xxwhichwecannowcallthepopulationvariance.ThesampleisdefinedXnTheexpression(n–1)isknownasthedegreesofIfthesamplesizeisreasonablylarge,wecandividebyninsteadof(n–1).SX(thepositivesquarerootofS2),iscalledthesamplex31–2S2 nSampleMeanandNnXXμ NX nNnXXSampleMeanandNnXXμ NX nNnXXi22iσ2 N nσs32–CovariancemeasureshowonerandomvariableCovariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.CovariancerangesfromnegativeinfinitytopositivePropertiesofIfXandYareindependentrandomvariables,theircovarianceisCovX,XEXEXXEX2XCov(abX,cdY)bdCov(X,IfXandYareNOTindependent,σ2(X±Y)=σ2(X)+σ2(Y)±33–CovX,YEXEXYEYEXYEXEYCorrelationCorrelationPropertiesofCorrelationCorrelationCorrelationPropertiesofCorrelationCorrelationhasnounits,rangesfrom-1toCorrelationmeasuresthelinearrelationshipbetweentwoIftwovariablesareindependent,theircovarianceiszero,therefore,thecorrelationcoefficientwillbezero.Theconverse,however,isnottrue.Forexample,Y=X2.Variancesofcorrelatedσ2(X±Y)=σ2(X)+σ2(Y)±34–CovρXY σ Correlationperfectpositivecorrelationr=perfectpositivecorrelationr=perfectpositivecorrelationr=perfectpositivecorrelationr=-0.7perfectpositivecorrelationr=-135–Correlationr=perfectpositive0<r<positivelinearCorrelationperfectpositivecorrelationr=perfectpositivecorrelationr=perfectpositivecorrelationr=perfectpositivecorrelationr=-0.7perfectpositivecorrelationr=-135–Correlationr=perfectpositive0<r<positivelinearr=nolinear-1<r<negativelinearr=-perfectnegativeMeasuresof36–wMeasuresof36–wmarketvalueofinvestmentinasset marketvalueofthe σ2(R)wwCov(R,R NE(RP)wiE(Ri)w1E(R1)w2E(R2) wnE(RNAmeasureofasymmetryofaPDF–thethirdSymmetricalandnonsymmetricalPositivelyAmeasureofasymmetryofaPDF–thethirdSymmetricalandnonsymmetricalPositivelyskewed(rightskewed)andnegativelyskewed(leftNegative-MeanMedianMean=Median=ModeMedianPositiveskewed:Mode<median<mean,havingarightfatNegativeskewed:Mode>media>mean,havingaleftfat37–EXμ3 thirdmomentaboutmeanS σ3x cubeofstandarddeviationxAmeasureoftallnessorflatnessofaPDF–thefourthForAmeasureoftallnessorflatnessofaPDF–thefourthForanormaldistribution,theKvalueisExcesskurtosis=kurtosis–38–>=<Excess>=<(assumingsamefatthinEXμ4 K 2EXμ2 squareofsecond Fat39–AFat39–AleptokurticdistributionhasmorefrequentextremelylargefromthemeanthananormalCoskewnessandCoskewnessandThethirdcrosscentralmomentisCoskewnessandCoskewnessandThethirdcrosscentralmomentisknownasThefourthcrosscentralmomentisknownas40–CoskewnessandAssumefourseriesoffundreturns(A、B、C、D)wherethemean,standarddeviation,skew,andkurtosisareallthesame,butonlytheorderofreturnsisdifferent:Thetwoportfolios(A+andC+D)havethesamemeanstandarddeviation,buttheskewsoftheportfoliosare41–A+C+CoskewnessandAssumefourseriesoffundreturns(A、B、C、D)wherethemean,standarddeviation,skew,andkurtosisareallthesame,butonlytheorderofreturnsisdifferent:Thetwoportfolios(A+andC+D)havethesamemeanstandarddeviation,buttheskewsoftheportfoliosare41–A+C+1234567ABCD1234567CoskewnessandScatterplotsshowtheCoskewnessandScatterplotsshowthedifferencebetweenBversusAandDversusAandB:theirbestpositivereturnsoccurduringthesametimeperiod,buttheirworstnegativereturnsoccurindifferentperiods.Thiscausesthedistributionofpointstobeskewedtowardthetop-rightofthechart.CandD:theirworstnegativereturnsoccurinthesameperiod,buttheirbestpositivereturnsoccurindifferentperiods.Inthesecondchart,thepointsareskewedtowardthebottom-leftofthe42–CoskewnessandThereasontheabovechartslookdifferentorthereasonthe CoskewnessandThereasontheabovechartslookdifferentorthereasonthe betweentheportfoliosisdifferent.Thenontrivialcoskewnessoftwovariables:andForandThenontrivialcokurtosisoftwovariablesKXXXY、For43–EXμ3Yμ Y σ3 EXμ2Yμ Y σ2 AandCand--CentralThek-thmomentofXisCentralThek-thmomentofXisdefinedmKEXKIfk=1,thenm1=E[X],itistheCentralThek-thcentralmomentofXisdefined EXKKCentralmomentsaremeasuredrelativetotheIfk=1,thefirstcentralmomentisequaltoIfk=2,thesecondcentralmomentistheIfk=3,thenthethirdcentralmomentdividedbythecubestandarddeviationistheskewness.ofIfk=4,thenthefourthmomentdividedbythesquareoftheisthe44–Chebyshev’sForanysetofobservations(samplesorpopulation),theproportionofthevaluesthatliewithinkstandarddeviationsofthemeanisatleast1–1/k2,k>1.Thisrelationshipappliesregardlessoftheshapeoftheofthemean45–Chebyshev’sForanysetofobservations(samplesorpopulation),theproportionofthevaluesthatliewithinkstandarddeviationsofthemeanisatleast1–1/k2,k>1.Thisrelationshipappliesregardlessoftheshapeoftheofthemean45–11113 1118 111115 234PXμkσ11,kExampleSupposethatExampleSupposethatAandBarerandomvariables,eachfollowsastandardnormaldistribution,andthecovariancebetweenAandBis0.35.Whatisthevarianceof(3A+46–ExampleGiventhatxExampleGiventhatxandyarerandomvariables,anda,b,canddarewhichoneofthefollowingdefinitionsisE(ax+by+c)=aE(x)+bE(y)+c,ifxandyarecorrelated.σ2(ax+by+c)=σ2(ax+by)+c,ifxandyarecorrelated.Cov(ax+by,cx+dy)=acσ2(x)+bdσ2(y)+(ad+bc)Cov(x,y),ifxandyarecorrelated.σ2(x–y)=σ2(x+y)=σ2(x)+σ2(y),ifxandyare47–ExampleWhichoneofExampleWhichoneofthefollowingstatementsaboutthecorrelationcoefficientItalwaysrangesfrom-1toAcorrelationcoefficientofzeromeansthattworandomvariablesareItisameasureoflinearrelationshipbetweentworandomItcanbecalculatedbyscalingthecovariancebetweentworandom48–BasicBasicLinearLinearRegressionwithOneLinearRegressionwithMultipleSimulationEstimatingVolatilitiesand49–3.DiscreteProbabilityBernoulliBinomialDiscreteProbabilityBernoulliBinomialPoissonContinuousProbabilityContinuousUniformNormalTheStandardNormalLognormalOtherCommonlyusedProbabilityChi-SquaretFParametricandNonparametricMixture50–BinomialBernoulliP(X=1)=P(X=0)=1BinomialBernoulliP(X=1)=P(X=0)=1–BinomialTheprobabilityofxsuccessesinnExpectationsand51–Bernoullirandompp(1–Binomialrandomnp(1–pxPXxCxpx1pnx px1 x!nSomeImportantProbabilityTheCumulativeBinomialProbabilityDerivingIndividualProbabilitiesfromCumulativeForexample,P(3)=F(3)–F(2)=0.813–0.500=52–PXFxFxFxPXxSomeImportantProbabilityTheCumulativeBinomialProbabilityDerivingIndividualProbabilitiesfromCumulativeForexample,P(3)=F(3)–F(2)=0.813–0.500=52–PXFxFxFxPXx allix012345SomeImportantProbabilityTheBinomialDistribution–p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandasp=53–xxSomeImportantProbabilityTheBinomialDistribution–p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandasp=53–xxxxxxxxPoissonPoissonWhentherearealargenumberoftrialsbutasmallsuccess,BinomialcalculationsbecomebabilitynDistributionbecomesPoissonPoissonWhentherearealargenumberoftrialsbutasmallsuccess,BinomialcalculationsbecomebabilitynDistributionbecomesthePoissonXreferstothenumberofsuccessperλindicatestherateofoccurrenceoftherandomevents;i.e.,ittellsushowmanyeventsoccuronaverageperunitoftime.Thenumberoffishcaughtinaday;thenumberofpotholesonakmstretchofroad;thenumberofpersonsappearedinashoppingmall;thenumberofphonecallsinaday.54– pkPXk λk!PoissonE(X)=D(X)=afurtherPoissonE(X)=D(X)=afurthervariablewithmeanequaltothesumoftheindividual Distributionasngoestoinfinityandpgoestozero,whilenp=λremainsfixed.Inaddition,whenλislargethePoissonDistributioniswellapproximatedbytheNormalDistributionwithmeanandvarianceofλ,throughthecentrallimit55–SomeImportantProbabilityAcompanyreceivesSomeImportantProbabilityAcompanyreceivesthreecomplaintsperdayonaverage.Whattheprobabilityofreceivingmorethanonecomplaintλ=a“morethanone”meansthatk=2or3or4or…P(‘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=56–ContinuousUniformProbabilitydensityCumulativedistributionabContinuousUniformProbabilitydensityCumulativedistributionab57– forxaF(x)xa foraxbb forx f(x)ba ContinuousUniformE(X)=ContinuousUniformE(X)=(a+D(X)=(b–Foralla≤x1<x2≤b,weTherandomvariableXwithdensityfunctionf(x)=k/3for2≤x≤8,and0otherwise.Calculateitsmean.58– xPxXx fxdx bNormalNormalAsnincreases,thebinomialdistributionapproachesNormalThenormalcurveisNormalNormalAsnincreases,thebinomialdistributionapproachesNormalThenormalcurveisThetwohalvesareextendsto-∞.extendsto+∞.Themean,median,andmodeareX~N(µ,σ²),fullydescribedbyitstwoparametersµandBell-shaped,symmetricaldistribution:skewness=0;kurtosis=Alinearcombination(function)oftwo(ormore)normallydistributionrandomvariablesisitselfnormallydistributed.Thetailsgetthinandgotozerobutextendinfinitely,59– fx e-22x-NormalTheconfidenceApproximately68%ofallobservationsfallintheintervalNormalTheconfidenceApproximately68%ofallobservationsfallintheintervalApproximately90%ofallobservationsfallintheintervalApproximately95%ofallobservationsfallintheintervalApproximately99%ofallobservationsfallintheinterval---60–TheStandardNormalThestandardnormalN(0,1)orStandardization:ifX~N(µ,σ²),TheStandardNormalThestandardnormalN(0,1)orStandardization:ifX~N(µ,σ²),Z-Example:X~N(70,9),computetheprobabilityofX≤ZXμ64.1270-σ3P(Z≤-1.96)=Question1:computetheprobabilityofX≥Question2:computetheprobabilityof64.12≤X≤61–ZXμσTheStandardNormalTheStandardNormal62–NormalDistribution–ExampleLetZNormalDistribution–ExampleLetZbeastandardnormalrandomvariable,andeventXisdefinedhappenifeitherZtakesavaluebetween-0.5and+0.5orZtakesvaluegreaterthen1.5.WhatistheprobabilityofXhappeningN(0.5)=0.6915andN(-1.5)=0.0668,wheredistributionfunctionofastandardnormal63–NormalDistribution–ExampleWhichofthefollowingstatementaboutnormalNormalDistribution–ExampleWhichofthefollowingstatementaboutnormaldistributionisKurtosisequalsthree.Skewnessequalsone.Theentiredistributioncanbecharacterizedbytwomoments,meanandvariance.Thenormaldensityfunctionhasthefollowingexpx211f(x)64–NormalDistribution–ExampleNormalDistribution–ExampleWhichtypeofdistributionproducesthelowestprobabilityforatoexceedaspecialextremevaluewhichisgreaterthanassumingthedistributionallhavethesamemeanandAleptokurticdistributionwithakurtosisof4.Aleptokurticdistributionwithakurtosisof8.Anormaldistribution.65–NormalDistribution–ExampleNormalDistribution–ExampleA$50millionprudentfund(PF)ismergedwitha$200millionaggressivefund(AF).ThereturnofPF~N(0.03,0.072)andthereturnofAF~N(0.07,0.152).Seniormanageraskedyoutoestimatethelikelihoodthatthereturnsofthecombinedportfoliowillexceed26%.Assumingthereturnsareindependent,whatistheprobabilitythatthereturnwillexceed66–LognormalLognormalTheBlack-ScholesModelassumesthatthepriceoftheunderlyingLognormalLognormalTheBlack-ScholesModelassumesthatthepriceoftheunderlyingassetislognormallydistributed.IflnXisnormal,thenXislognormal;ifavariableislognormal,itsnaturallogisnormal.ItisusefulformodelingassetpriceswhichnevertakenegativeRightBoundedfrombelowby67–lnX~Nμ,σ2EXexpμ1σ2 DXexp2μ2σ2exp2μσ2f(x) exp1lnx2,x0x22 Chi-SquareChi-Square(2)Chi-SquarecomputedChi-SquareChi-Square(2)Chi-Squarecomputedn–1df=68–Z2Z2Z2 Z2~ (n 0Chi-SquareTheChi-SquaredistributiontakeonlypositivevalueandrangesfromChi-SquareTheChi-Squaredistributiontakeonlypositivevalueandrangesfrom0toinfinity(afterall,itisthedistributionofasquaredquantity).TheChi-Squaredistributionisapositiveskeweddistribution,degreeoftheskewnessdependingontheForcomparativelyfewd.f.thedistributionishighlyskewedto increasinglysymmetricalandapproachesthenormalE(X)=k,D(X)=2k,wherekistheIfZ1andZ2aretwoindependentChi-Squarevariableswithk1andk2d.f.,thentheirsum(Z1+Z2)isalsoaChi-Squarevariablewithd.f.=(k1+69–ttDistribution(student’stX~N0,1,bothRecallthat,Zσ2areXxσX/SupposeweonlyknowttDistribution(student’stX~N0,1,bothRecallthat,Zσ2areXxσX/Supposeweonlyknowandbyits(sample)Xx2Xi,weobtainanewXnExplainthed.f.(degreesof(andhenceSx),wemustfirstcomputeXBeforewecomputethexButsinceweusethesamesampletocomputeX,wehave(n-1),n,independentobservationstocomputeS2,sotospeak,welosex70–tX ~tS n xtThemeanoftdistributioniszero,anditsvariancen/(n–tThemeanoftdistributioniszero,anditsvariancen/(n–tstandardnormaldistribution,sotdistributionisflatterthanthentapproachesthevarianceofthestandardnormaldistribution,71–F-F-IfU1andU2aretwoindependentChi-Squareddistributionswithandk2degreesoffreedom,respectively,thenX:followsanFF-F-IfU1andU2aretwoindependentChi-Squareddistributionswithandk2degreesoffreedom,respectively,thenX:followsanF-distributionwithparametersk1andAsd.f.increase,theF-DistributionapproachesNormalIfXisahaswithatF-Distributionwithkk72–X2~F1,XU1 ~Fk,k F-F-F0TheFF-F-F0TheFdistributionforvariousSkewedtotherightandalsorangesbetween0andApproachestheNormalDistributionask1andk2,thed.f.73–ExampleTheannualmarginalprobabilityofdefaultofExampleTheannualmarginalprobabilityofdefaultofabondis15%inyear120%inyear2.Whatisthedefault)totheendoftwoAnswer:Probability(nodefault)=(1–15%)×(1–20%)=74–ExampleOnamultiplechoiceexamExampleOnamultiplechoiceexamwithfourchoicesforeachofsixquestions,whatistheprobabilitythatastudentgetslessthantwoquestionssimplybyAnswer:p(0)==p(1)==Theprobabilityofgettinglessthantwoquestionscorrectisp(0)+=75–ExampleAcallcenterreceivesanaverageoftwophoneExampleAcallcenterreceivesanaverageoftwophonecallsperhour.probabilitythattheywillreceive20callsinan8-hourdayisclosestAnswer:thisquestion,wefirstneedtorealizethatthenumberofphonecallsinan8-hourdayis16.Usingthedistribution,wesolvefortheprobabilitythatXwillbe1620e-P(X20)76–ExampleIfwesaythatcommoditypricefollowalognormaldistribution,ExampleIfwesaythatcommoditypricefollowalognormaldistribution,wethatoverThenaturallogarithmofthepriceisnormallyThechangeinthepriceisnormallydistributedoverThereciprocalofthepriceisnormallyAnswer:Arandomvariablehasalognormaldistributionifitslogarithmisitselfnormallydistributed.77–LinearLinearLinearRegressionwithOneLinearRegressionwithMultipleSimulationEstimatingVolatilitiesand78–4.HypothesisTestsandConfidenceSamplingandPointSamplingandPointEstimation、ConfidenceIntervalBestLinearUnbiasedEstimatorHypothesisThebasisofTheapplicationofTestofSinglePopulationMeanTestofVariances79–SampleandSamplingandimportantcharacteristicslargedataInferentialstatistics:Makeforecasts,estimates,orjudgmentsaboutalargesetofdataonthebasisSampleandSamplingandimportantcharacteristicslargedataInferentialstatistics:Makeforecasts,estimates,orjudgmentsaboutalargesetofdataonthebasisofthestatisticalcharacteristicsofasmallerset(a80–sampleStatisticalInference:EstimationandHypothesisSamplingandSimpleStatisticalInference:EstimationandHypothesisSamplingandSimplerandomStratifiedrandomsampling:toseparatethepopulationintosmallergroupsbasedononeormoredistinguishingcharacteristics.Stratumandcells=M×N.Samplingsamplingerrorofthemean=samplemean–populationTheaa81–TheCentralLimitTheCentralLimitTheoremX1,,Xnisarandomsamplefromanypopulation(i.e.,distribution)TheCentralLimitTheCentralLimitTheoremX1,,Xnisarandomsamplefromanypopulation(i.e.,distribution)withmean,thesampleXtendstoastheXxnormallydistributedwithμand/nXxOfcourse,ifhappentobefromthenormalpopulation,samplemeanfollowsthenormaldistributionregardlessoftheStandardError(SE)ofmeanknown.Instead,weusethestandarddeviationofthe82–SEXnStatisticalInference:EstimationandHypothesisEstimationStatisticalInference:EstimationandHypothesisEstimationandHypothesisTesting:TwinBranchesOfStatisticalPricetoearning(P/E)ratiosof28companiesontheNewYorkstockexchangeMean=Variance=Standarddeviation=83–BestLinearUnbiasedEstimatorPropertiesofpointThemeanoftheestimatorscoincideswiththetrueparametere.g.BestLinearUnbiasedEstimatorPropertiesofpointThemeanoftheestimatorscoincideswiththetrueparametere.g.E(X)AnunbiasedestimatorisalsoefficientifthevarianceofitsdistributionissmallerthanalltheotherunbiasedestimatorsparameteryouaretryingtoX~n)Theaccuracyoftheparameterincreases(seethestandardincreasesastheXnn84–BestLinearUnbiasedBestLinearUnbiasedEstimatorAnotherpropertyofapointestimateislinearity.Apointestimateshouldbealinearestimator(i.e.,itcanbeusedasalinearfunctionofthesampledata).Iftheestimatoristhebestavailable(i.e.,hastheminimumvariance),exhibitslinearity,andisunbiased,itissaidtobethebestlinearunbiasedestimator85–PointEstimationandConfidenceIntervalPointUsingasinglevaluetoPointEstimationandConfidenceIntervalPointUsingasinglevaluetoConfidenceIntervalLevelofsignificanceDegreeofconfidence(1–0Thetdistributionfor27ConfidenceInterval=[PointEstimate+/-(reliabilityfactor)×86–X S2 ConfidenceConfidenceIntervalThepopulationhasanormaldistributionwithaknownConfidencePointReliabilityStandardThepopulationhasConfidenceConfidenceIntervalThepopulationhasanormaldistributionwithaknownConfidencePointReliabilityStandardThepopulationhasanormaldistributionwithaunknownConfidence87–Whensamplingformsmallsample(n<smallsample(n≥NormaldistributionwithknownNormaldistributionwithunknownt-Statisticorz-NonnormaldistributionwithknownnotNonnormaldistributionwithunknownnott-Statisticorz-X nXσ nStatisticalInference:EstimationandHypothesisExample:ConfidenceIntervaltvaluetvalueXt~t(nCriticalStatisticalInference:EstimationandHypothesisExample:ConfidenceIntervaltvaluetvalueXt~t(nCritical0Thetdistributionfor27P(-2.052t2.052)ObtainaRandomPX nXX nImportantpoint:oneshouldsaythattheprobabilityis0.95thatintervalcontainsthetrue88–StatisticalInference:EstimationandHypothesisUsingsampleestimator,obtainingaconfidenceReturningtoourP/Eexample,weStatisticalInference:EstimationandHypothesisUsingsampleestimator,obtainingaconfidenceReturningtoourP/Eexample,wehaven=28,X23.25,andSX9.49.Weobtain19.57≤for≤26.93asthe95%confidenceSSXXXX n nX89–HypothesisStepStepStepStepStep90–decisionTakeasample,HypothesisStepStepStepStepStep90–decisionTakeasample,DonotSelectalevelofFormulateIdentifythetestStatenullandHypothesisStatisticalassessmentofastatementorHypothesisStatisticalassessmentofastatementoridearegardingapopulationThenullhypothesis(H0)andalternativehypothesisOne-tailedtestvs.Two-tailedOne-tailedTwo-tailedCriticalThedistributionofteststatistic(z,t,2,SignificancelevelOne-tailedortwo-tailed91–H0:μ= Ha:μ≠H0:μ≤ Ha:μ>H0