v3 frm一級(jí)基礎(chǔ)定量分析教育

么崢老師簡(jiǎn)教育背景：浙江大學(xué)數(shù)學(xué)學(xué)士么崢老師簡(jiǎn)教育背景：浙江大學(xué)數(shù)學(xué)學(xué)士，浙江大學(xué)金融學(xué)碩的BaselII與BaselIII定量測(cè)算；撰寫新資本協(xié)議實(shí)施的相關(guān)政策與辦法；曾多次組織浙江省、江講授課程：FRM一級(jí)數(shù)量分析，金融市場(chǎng)與產(chǎn)品，估值與風(fēng)險(xiǎn)模型；FRM二級(jí)操作風(fēng)險(xiǎn)，巴箱：新TopicWeightingsinFRMPartStudySessionFoundationsofRiskStudySessionFinancialMarketsTopicWeightingsinFRMPartStudySessionFoundationsofRiskStudySessionFinancialMarketsandStudySessionValuationandRiskStudySession Quantitative Session LinearRegressionwithOneLinearRegressionwithOneModelingcyclesModelingandForecastingCorrelationsand BasisRandomexperimentOutcome&EventRandomvariable????Probability&Probabilityalgorithm概率運(yùn)ConceptofProbabilityMultiplicationrule、Additionrule BasisRandomexperimentOutcome&EventRandomvariable????Probability&Probabilityalgorithm概率運(yùn)ConceptofProbabilityMultiplicationrule、AdditionruleProbabilityFunction&CumulativeDistributionFunctionrandomvariables????Randomexperiment隨機(jī)試驗(yàn)AnobservationormeasurementprocesswithmultiplebutuncertainOutcome結(jié)果Theresultofasingletrial.Forexample,ifwerolltwodices,anoutcomemightbe3and4;adifferentoutcomemightbe5and2.Event事件TheresultRandomexperiment隨機(jī)試驗(yàn)AnobservationormeasurementprocesswithmultiplebutuncertainOutcome結(jié)果Theresultofasingletrial.Forexample,ifwerolltwodices,anoutcomemightbe3and4;adifferentoutcomemightbe5and2.Event事件Theresultthatreflectsnone,one,ormoreoutcomesinthesamplespace.Eventscanbesimpleorcompound.Aneventisasubsetofthesamplespace.Ifwerolltwodices,anexampleofaneventmightberolling7inMutuallyexclusiveevents互斥事件):Eventsthatcannotbothhappenatthesametime.Exhaustiveevents完備事件ThoseincludeallpossibleProbabilityofanTwodefiningpropertiesof0≤P(E)≤IfE1,E2ProbabilityofanTwodefiningpropertiesof0≤P(E)≤IfE1,E2,……,Enismutuallyexclusiveandexhaustive,then:P(E1)+P(E2)+……+P(En)=1 APA=totalnumberofoutcomesJointTheprobabilitythattherandomvariables(inthiscase,bothrandomvariables)takeoncertainvaluessimultaneously,P(AB).UnconditionalProbability邊際概率a.k.aJointTheprobabilitythattherandomvariables(inthiscase,bothrandomvariables)takeoncertainvaluessimultaneously,P(AB).UnconditionalProbability邊際概率a.k.amarginalTheexpectedvalueofthevariablewithoutanyrestrictions(orlackinganypriorinformation),P(A).informationorsomerestriction(e.g.,thevalueofacorrelatedvariable).TheconditionalexpectationofB,conditionalonA,isgivenbyUnconditionalprobability:P(A),P(B|A)=Unconditionalprobability:P(A),P(B|A)=P(AB);P(A)>P(A|B)=P(AB);P(B)>0Jointprobability:MultiplicationP(AB)=P(A|B)×P(B)Jointprobability: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)+TheoccurrenceofAhasnoinfluenceTheoccurrenceofAhasnoinfluenceofontheoccurrenceofP(A|B)=P(A)orP(B|A)=P(AB)=P(A)×P(AorB)=P(A)+P(B)–IndependenceandMutuallyExclusivearequiteIfexclusive,mustnotCauseexclusivemeansifAoccur,Bcannotoccur,AinfluentsP(AB)=P(A)×IfaneventAmustresultinoneofthemutuallyexclusiveeventsA1,A2,A3,……,An,then(1)AA=FIfaneventAmustresultinoneofthemutuallyexclusiveeventsA1,A2,A3,……,An,then(1)AA=F(i≠ n=W PA)=PA1PAA1)+PA2PAA2)+...+PAnPAAnX:Company’schoicedefault–{0,1}ANotBNotX:Company’schoicedefault–{0,1}ANotBNotCNotBayes’Prior一個(gè)人有病的概率是10%，沒病的概率是90。在有病的情況下5%95%Bayes’Prior一個(gè)人有病的概率是10%，沒病的概率是90。在有病的情況下5%95%PA|B)=PB|A·PA)PB)人P(A|B)= 99%·10% 人P(A|B)= 99%·10% =68.75%99%·10%+5%·90%Describetheprobabilitiesofallthepossibleoutcomesrandomvariable.DiscreteandcontinuousrandomforDiscreterandomvariables:Describetheprobabilitiesofallthepossibleoutcomesrandomvariable.DiscreteandcontinuousrandomforDiscreterandomvariables:thenumberofpossibleoutcomesameasurableandpositive infinite,eveniflowerandupperboundsP(x)=0eventhoughxcanP(x1<X<Probabilityfunction:p(x)=P(X=FordiscreterandomProbabilityfunction:p(x)=P(X=Fordiscreterandom0≤p(x)≤Σp(x)=Probabilitydensityfunction(p.d.f):ForcontinuousrandomvariableCumulativeprobabilityfunction(c.p.f):xF(x)=F(x)=P(X≤RandomVariablesandTheirProbabilityProbabilityDistributionofaDiscreteRandomProbabilityMassFunction(PMF)orProbabilityFunctionPropertiesRandomVariablesandTheirProbabilityProbabilityDistributionofaDiscreteRandomProbabilityMassFunction(PMF)orProbabilityFunctionPropertiesoftheForn=3p= Binomial:n=3,p=123 X033.f(xi)=x2.0￡f(xi)￡1.f(X=xi)=0,x?f(X=xi)=P(X=xi),i=ProbabilityDistributionofaContinuousRandomProbabilitythatheightProbabilitydensityfunctionbetween60ProbabilityDistributionofaContinuousRandomProbabilitythatheightProbabilitydensityfunctionbetween60and68X0HeightinAPDFhasthefollowingThetotalareaunderthecurvef(x)isP(x1<X<x2)istheareaunderthecurvebetweenx1andP(x1￡X￡x2)=P(x1<X￡x2)=P(x1￡X<x2)=P(x1<X<x2P(x1<X<x2)=CumulativeDistributionFunction10abx0abxP(a≤X≤CumulativeDistributionFunction10abx0abxP(a≤X≤b)=Areaunderf(x)betweenaandb=F(b)–P(a≤X≤b)=F(b)–F(X)=P(X￡P(guān)ropertiesofF(-∞)=0andF(+∞)=PropertiesofF(-∞)=0andF(+∞)=F(X)isanon-decreasingfunctionsuchthatifx2P(X≥k)=1–P(x1≤X≤x2)=F(x2)–>thenF(x2)WetakeXfrom1or2withthesameprobability.WeWetakeXfrom1or2withthesameprobability.WetakeYfrom[1,X]withthesameprobability.Definition:f(X,Y)=P(X=xandPropertiesofthebivariateorjointprobabilitymassfunctionf(X,Y)≥0forallpairsofXandY.Thisisbecauseallprobabilitiesare∑∑f(X,Y)YX1212MarginalprobabilityDefinitionofmarginalprobabilityf(Y)=f(X,Y)forallxf(X)=MarginalprobabilityDefinitionofmarginalprobabilityf(Y)=f(X,Y)forallxf(X)=f(X,Y)forallyMarginalprobabilitydistributionofXandValueofValueof1122DefinitionofStatisticalIndependence:f(X,Y)=XY123123DefinitionofStatisticalIndependence:f(X,Y)=XY1231231ThejointprobabilitydistributionofrandomvariablesXandYisgivenbyf(x,y)=kxyforx=1,2,3,yThejointprobabilitydistributionofrandomvariablesXandYisgivenbyf(x,y)=kxyforx=1,2,3,y=1,2,3andkisapositiveconstant,whatistheprobabilitythatX+Ywillexceed5?CannotbeHalfofthemortgagesinaportfolioareconsideredsubprime.Theprincipalbalanceofhalfofthesubprimemortgagesandone-quarterofthenon-subprimemortgagesexceedsthevalueofthepropertyusedascollateral.Ifyourandomlyselectamortgagefromtheportfolioforreviewanditsprinciplebalanceexceedsthevalueofthecollateral,whatistheprobabilitythatitisasubprimemortgage?Halfofthemortgagesinaportfolioareconsideredsubprime.Theprincipalbalanceofhalfofthesubprimemortgagesandone-quarterofthenon-subprimemortgagesexceedsthevalueofthepropertyusedascollateral.Ifyourandomlyselectamortgagefromtheportfolioforreviewanditsprinciplebalanceexceedsthevalueofthecollateral,whatistheprobabilitythatitisasubprimemortgage?BasicBasicCoskewnessCoskewnessandCokurtosisCentralMomentAmeasureofcentraltendency–thefirstPropertiesAmeasureofcentraltendency–thefirstPropertiesofExpectedIfbisaconstant,E(b)=Ifaisaconstant,E(aX)=Ifaandbareconstants,thenE(aX+b)=aE(X)+E(b)=aE(X)+E(X2)≠E(X+Y)=E(X)+Ingeneral,E(XY)≠E(X)E(Y);IfXandYareindependentrandomvariables,thenE(XY)=E(X)E(Y).E(X)=E(X)=P(xi)xi=P(x1)x1+P(x2)x2+...+Ameasureofdispersion–thesecondAboveformulaisthedefinitionofvariance,Ameasureofdispersion–thesecondAboveformulaisthedefinitionofvariance,weusethefollowingTocomputeMeasureshownoisyorunpredictablethatrandomvariableThepositivesquarerootσ2,,isknownasthexdeviation,alsocalled2EX2EXEXPropertiesofThevarianceofaconstantiszero.Bydefinition,aconstanthasnovariability.Ifaisconstant,then:σ2(aX)=PropertiesofThevarianceofaconstantiszero.Bydefinition,aconstanthasnovariability.Ifaisconstant,then:σ2(aX)=Ifbisaconstant,then:σ2(X+b)=Ifaandbareconstant,then:σ2(aX+b)=IfXandYaretwoindependentrandomvariables,IfXandYareindependentrandomvariablesandaandbareconstants,thenσ2(aX+bY)=a2σ2(X)+b2σ2(Y).Forcomputationalconvenience,wecanget:σ2(X)=E(X2)–[E(X)]2,thatEXxσ2(X–Y)=σ2(X)+σ2(X+Y)=σ2(X)+SampleSampleThesamplemeanofarandomvariable,X,isdefined=TheSampleSampleThesamplemeanofarandomvariable,X,isdefined=ThesamplemeanisknownasanestimatorofE(X),whichwenowcallthepopulationAnestimateofthetakenbyani=1SampleSampleThesamplevariance,denotedbyS2whichisanestimatorofσ2SampleSampleThesamplevariance,denotedbyS2whichisanestimatorofσ2xxwhichwecannowcallthepopulationvariance.ThevarianceisdefinedTheexpression(n–1)isknownasthedegreesofIfthesamplesizeisreasonablylarge,wecandividebyninsteadof(n–1). (thepositivesquarerootofS2),iscalledthesampleXx XiS2=x n-SampleMeanandNXiμ= NnXiX=SampleMeanandNXiμ= NnXiX= n σ2= N X-2is2= n-σsCovariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.CovariancerangesfromnegativeinfinityCovariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.CovariancerangesfromnegativeinfinitytopositivePropertiesofIfXandYareindependentrandomvariables,theircovarianceisX,EXXX)Cov(a+bX,c+dY)=b·d·Cov(X,IfXandYareNOTindependent,σ2(X±Y)=σ2(X)+σ2(Y)±CovX,Y)=E(X-EXY-=EXY-EXEY)CorrelationPropertiesofCorrelationCorrelationhasnounits,rangesfrom-1to randomCorrelationPropertiesofCorrelationCorrelationhasnounits,rangesfrom-1to randomtherefore,thecorrelationcoefficientwillbezero.Theconverse,however,isnottrue.Forexample,Y=X2.Variancesofcorrelatedσ2(X±Y)=σ2(X)+σ2(Y)± X,Y) σ perfectpositivecorrelationr=+1perfectpositivecorrelationr=0.8perfectpositivecorrelationr=perfectpositivecorrelationr=-0.7perfectpositivecorrelationr=-1perfectpositivecorrelationr=+1perfectpositivecorrelationr=0.8perfectpositivecorrelationr=perfectpositivecorrelationr=-0.7perfectpositivecorrelationr=-1Correlationr=perfectpositive0<r<positivelinearr=nolinear-1<r<negativelinearr=-perfectnegativeMeasuresofw=marketvalueofinvestmentinasset marketvalueofMeasuresofw=marketvalueofinvestmentinasset marketvalueofthe σ2(RP)=wiwjCov(Ri,Rj NE(RP)=wiE(Ri)=w1E(R1)+w2E(R2)+L+wnE(RNAmeasureofasymmetryofaPDF–thethirdPositivelyskewedAmeasureofasymmetryofaPDF–thethirdPositivelyskewed(rightskewed)andnegativelyskewed(leftMean=Median=ModeMedianMedianPositiveskewed:Mode<median<mean,havingarightfattailNegativeskewed:Mode>media>mean,havingaleftfattailEX-μ3 thirdmomentaboutmeanS σ3 =cubeofstandardxAmeasureoftallnessorflatnessofaPDF–thefourthForanormaldistribution,theKvalueisAmeasureoftallnessorflatnessofaPDF–thefourthForanormaldistribution,theKvalueisExcesskurtosis=kurtosis–>=<Excess>=<(assumingsamevariance)fatthinEX-μ4 K= 2=2 squareofsecond FatAleptokurticdistributionhasmorefrequentextremelylargeFatAleptokurticdistributionhasmorefrequentextremelylargefromthemeanthananormalCoskewnessandCoskewnessandThethirdcrosscentralmomentisCoskewnessandCoskewnessandThethirdcrosscentralmomentisknownasThefourthcrosscentralmomentisknownasCoskewnessandAssumefourseriesoffundreturns(A、B、C、D)wherethemean,standarddeviation,skew,andkurtosisareallthesame,butonlytheorderofreturnsisThetwoportfolios(A+BandC+D)havethesamemeanandstandarddeviation,buttheskewsoftheportfoliosaredifferent.ACoskewnessandAssumefourseriesoffundreturns(A、B、C、D)wherethemean,standarddeviation,skew,andkurtosisareallthesame,butonlytheorderofreturnsisThetwoportfolios(A+BandC+D)havethesamemeanandstandarddeviation,buttheskewsoftheportfoliosaredifferent.A+C+1234567ABCD1234567CoskewnessandScatterplotsshowthedifferencebetweenBversusAandCoskewnessandScatterplotsshowthedifferencebetweenBversusAandDversusAandB:theirbestpositivereturnsoccurduringthesametimeperiod,buttheirworstnegativereturnsoccurindifferentperiods.Thiscausesthedistributionofpointstobeskewedtowardthetop-rightoftheCandD:theirworstnegativereturnsoccurinthesameperiod,buttheirbestpositivereturnsoccurindifferentperiods.Inthesecondchart,thepointsareskewedtowardthebottom-leftoftheCoskewnessandThereasontheabovechartslookdifferentorthereasonthe coskewnessbetweentheportfoliosCoskewnessandThereasontheabovechartslookdifferentorthereasonthe coskewnessbetweentheportfoliosisThenontrivialcoskewnessoftwovariables:SXXYandForThenontrivialcokurtosisoftwovariables:KXXXY、andFor3 = Y σ3 2Y = Y σ2 AandCandCentralThek-thmomentofXisdefinedEXKCentralThek-thmomentofXisdefinedEXKIfk=1,thenm1=E[X],itistheCentralThek-thcentralmomentofXisdefinedas:mKKCentralmomentsaremeasuredrelativetotheIfk=1,thefirstcentralmomentisequaltoIfk=2,thesecondcentralmomentistheIfk=3,thenthethirdcentralmomentdividedbythecubeofthestandarddeviationistheskewness.Ifk=4,thenthefourthcentralmomentdividedbythesquareofthevarianceistheportionofthevaluesthatliewithinkstandarddeviationsthemeanisatleast1–1/k2,kproportionofthevaluesthatliewithinkstandarddeviationsthemeanisatleast1–1/k2,k> ?1 =1-== ?1-1=1-1=8= 1 1 ?1 =1 = 234SupposethatAandBarerandomvariables,eachfollowsastandardnormaldistribution,andthecovariancebetweenAandBis0.35.Whatisthevarianceof(3A+2B)?SupposethatAandBarerandomvariables,eachfollowsastandardnormaldistribution,andthecovariancebetweenAandBis0.35.Whatisthevarianceof(3A+2B)?Giventhatxandyarerandomvariables,anda,b,cGiventhatxandyarerandomvariables,anda,b,canddareconstant,whichoneofthefollowingdefinitionsiswrong?E(ax+by+c)=aE(x)+bE(y)+c,ifxandyareσ2(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),ifxandyareWhichoneofthefollowingstatementsaboutthecorrelationcoefficientisfalse?ItalwaysWhichoneofthefollowingstatementsaboutthecorrelationcoefficientisfalse?Italwaysrangesfrom-1toAcorrelationcoefficientofzeromeansthattworandomvariablesareindependent.ItisameasureoflinearrelationshipbetweentworandomItcanbecalculatedbyscalingthecovariancebetweentworandomvariables.BernoulliDistributionBinomialDistributionPoissonDistributionContinuousProbabilityDistributionContinuousUniformDistributionNormalDistributionLognormalOtherCommonlyusedProbabilitytDistributionFBernoulliDistributionBinomialDistributionPoissonDistributionContinuousProbabilityDistributionContinuousUniformDistributionNormalDistributionLognormalOtherCommonlyusedProbabilitytDistributionFParametricandNonparametricMixture??????????BernoulliP(X=1)=P(X=0)=1–BinomialTheprobabilityofBernoulliP(X=1)=P(X=0)=1–BinomialTheprobabilityofxsuccessesinnExpectationsandBernoullirandompp(1–Binomialrandomnp(1–px)=PX1pn- 1pn- x!nx!SomeImportantProbabilityTheCumulativeBinomialProbabilityDerivingIndividualProbabilitiesfromCumulativeForexample,P(3)=F(3)–F(2)=0.813–0.500=X)=Fx-FxF(x)=P(X￡x)=Pi)alli￡xSomeImportantProbabilityTheCumulativeBinomialProbabilityDerivingIndividualProbabilitiesfromCumulativeForexample,P(3)=F(3)–F(2)=0.813–0.500=X)=Fx-FxF(x)=P(X￡x)=Pi)alli￡x012345SomeImportantProbabilityTheBinomialDistribution–p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandas=xxSomeImportantProbabilityTheBinomialDistribution–p=p=p=n=n=n=Binomialdistributionsbecomemoresymmetricasnincreasesandas=xxxxxxxxWhentherearealargenumberoftrialsbutasmallprobabilityofsuccess,Binomialcalculationsbecomeimpractical.p,Whentherearealargenumberoftrialsbutasmallprobabilityofsuccess,Binomialcalculationsbecomeimpractical.p,nDistributionbecomesthePoissonXreferstothenumberofsuccessperλindicatestherateofoccurrenceoftherandomevents;i.e.,ittellsushowmanyeventsoccuronaverageperunitoftime.Thenumberoffishcaughtinaday;thenumberofpotholesona1kmstretchofroad;thenumberofpersonsappearedinashoppingmall;thenumberofphonecallsinaday.kpk)=PX λ=k!E(X)=D(X)=ThesumofindependentPoissonvariablesisafurtherPoissonvariablewithmeanequaltothesumoftheindividualmeans.ThePoissonDistributionisthelimitingcaseE(X)=D(X)=ThesumofindependentPoissonvariablesisafurtherPoissonvariablewithmeanequaltothesumoftheindividualmeans.ThePoissonDistributionisthelimitingcaseoftheBinomialDistributionasngoestoinfinityandpgoestozero,whilenp=λ λ DistributioniswellapproximatedbytheNormalwithmeanandvarianceofλ,throughthecentrallimitSomeImportantProbabilityAcompanyreceivesthreecomplaintsperdayonaverage.SomeImportantProbabilityAcompanyreceivesthreecomplaintsperdayonaverage.istheprobabilityofreceivingmorethanonecomplaintparticularday?λ=“morethanone”meansthatk=2or3or4orP(‘morethanone’)=P(2)+P(3)+P(4)+P(‘morethanone’)=1–{P(0)+P(0)=e-3×30/0!=P(1)=e-3×31/1!=P(0)+P(1)=aP(‘morethanone’)=1–{P(0)+P(1)}=1–0.1992=Cumulativedistributionab forx￡aF(x)Cumulativedistributionab forx￡aF(x)=x-a fora<x<bb- forx?1 f(x)=b-aE(X)=(a+D(X)=(b–E(X)=(a+D(X)=(b–Foralla≤x1<x2≤b,weTherandomvariableXwithdensityfunctionf(x)=k/3for2≤≤8,and0otherwise.Calculateits x-Px1xdx b-1NormalAsnincreases,ThenormalcurveisThetwohalvesareNormalAsnincreases,ThenormalcurveisThetwohalvesareTheoretically,thecurveextendsto-∞.Theoretically,thecurveextendsto+∞.Themean,median,andmodearef ( X~N(μ,σ2),fullydescribedbyitstwoparametersμandBell-shaped,X~N(μ,σ2),fullydescribedbyitstwoparametersμandBell-shaped,symmetricaldistribution:skewness=0;kurtosis=AdistributionrandomvariablesisitselfnormallyThetailsgetthinandgotozerobutextendinfinitely,TheconfidenceApproximately68%ofallobservationsfallintheintervalApproximately90%ofallTheconfidenceApproximately68%ofallobservationsfallintheintervalApproximately90%ofallobservationsfallintheintervalApproximately95%ofallobservationsfallintheintervalApproximately99%ofallobservationsfallintheintervalmTheStandardNormalThestandardnormalN(0,1)orStandardization:ifX~N(μ,σ2),TheStandardNormalThestandardnormalN(0,1)orStandardization:ifX~N(μ,σ2),variousExample:X~N(70,9),computetheprobabilityofX≤Z=X-μ=64.12-70?-3σP(Z≤-1.96)=Question1:computetheprobabilityofX≥Question2:computetheprobabilityof64.12≤X≤Z=X-μ~N0,1σTheStandardNormalTheStandardNormalLetZbeastandardnormalrandomvariable,andeventXLetZbeastandardnormalrandomvariable,andeventXisdefinedtohappenifeitherZtakesavaluebetween-0.5and+0.5orZtakesanyvaluegreaterthen1.5.WhatistheprobabilityofeventXhappeningifN(0.5)=0.6915andN(-1.5)=0.0668,whereN(.)isthecumulativedistributionfunctionofastandardnormalvariable?WhichofthefollowingstatementaboutthenormaldistributionisnotKurtosisequalsthree.SkewnessequalsTheentiredistributioncanbeWhichofthefollowingstatementaboutthenormaldistributionisnotKurtosisequalsthree.SkewnessequalsTheentiredistributioncanbecharacterizedbytwomoments,meanandvariance.Thenormaldensityfunctionhasthefollowing11exp2f(x)x-2Whichtypeofdistributionproducesthelowestprobabilityforavariabletoexceedaspecialextremevaluewhichisgreaterthanthemean,assumingthedistributionallhavethesamemeanandvariance?Whichtypeofdistributionproducesthelowestprobabilityforavariabletoexceedaspecialextremevaluewhichisgreaterthanthemean,assumingthedistributionallhavethesamemeanandvariance?Aleptokurticdistributionwithakurtosisof4.Aleptokurticdistributionwithakurtosisof8.Anormaldistribution.AplatykurticA$50millionprudentfund(PF)ismergedwitha$200millionaggressivefund(AF).ThereturnofPF~N(0.03,0.072)andtheA$50millionprudentfund(PF)ismergedwitha$200millionaggressivefund(AF).ThereturnofPF~N(0.03,0.072)andthereturnofAF~N(0.07,0.152).Seniormanageraskedyoutoestimatethelikelihoodthatthereturnsofthecombinedportfoliowillexceed26%.Assumingthereturnsareindependent,whatistheprobabilitythatthereturnwillexceed26%?LognormalTheBlack-ScholesModelassumesthatthepriceoftheunderlyingassetislognormallydistributed.IflnXisnormal,thenXislognormal;ifaLognormalTheBlack-ScholesModelassumesthatthepriceoftheunderlyingassetislognormallydistributed.IflnXisnormal,thenXislognormal;ifavariableislognormal,itsnaturallogisnormal.ItisusefulformodelingassetpriceswhichnevertakenegativeRightBoundedfrombelowbyNμ,σ2E(X)=expμ+1σ2 exp2μexp2μf(x)= lnxm2x Chi-Square(c2)Chi-Squareteststatistic,c2,withn–Chi-Square(c2)Chi-Squareteststatistic,c2,withn–1degreesofcomputedas:freedom,df=Z2=Z2+Z2+L+Z2~ (n- =0TheChi-Squaredistributiontakeonlypositivevalueandrangesfrom0toinfinity(afterall,itisthedistributionofasquaredquantity).TheChi-Squaredistributionisapositiveskeweddistribution,TheChi-Squaredistributiontakeonlypositivevalueandrangesfrom0toinfinity(afterall,itisthedistributionofasquaredquantity).TheChi-Squaredistributionisapositiveskeweddistribution,thedegreeoftheskewnessdependingonthed.f.Forcomparativelyfewd.f.thedistributionishighlyskewedto increasinglysymmetricalandapproachesthenormalE(X)=k,D(X)=2k,wherekistheIfZ1andZ2aretwoindependentChi-Squarevariableswithk1andk2d.f.,thentheirsum(Z1+Z2)isalsoaChi-Squarevariablewithd.f.=(k1+k2).ttDistribution(student’stX-Recall,are~N0,1)ZxσX/Supposeweonlyknowandx2 X-ttDistribution(student’stX-Recall,are~N0,1)ZxσX/Supposeweonlyknowandx2 X-,weobtainanew 2Xn-Explainthed.f.(degreesofSBeforewecompute(and),wemustxxcomputeX.ButsinceweusethesamesampletocomputeXwehave(n-1),notn,independentobservationstocomputeS2xsotospeak,weloset=X- ~S/n tThemeanoftdistributioniszero,anditsvariancen/(n–ThevarianceoftdistributiontThemeanoftdistributioniszero,anditsvariancen/(n–Thevarianceoftdistributionislargerthanthevarianceofthestandardnormaldistribution,sotdistributionisflatterthanthentdistributiondistribution,namelyF-IfU1andU2aretwoindependentChi-Squareddistributionsk1andk2degreesF-IfU1andU2aretwoindependentChi-Squareddistributionsk1andk2degreesoffreedom,respectively,thenfollowsanF-distributionwithparametersk1andIfXisarandomvariablewithatdistributionwithkdegreesoffreedom,thenX2hasanF-Distributionwith1andkdegreesofF1,X=U1k1~Fk,k F-F0TheFdistributionforvariousSkewedtotherightandalsorangesbetween0F-F0TheFdistributionforvariousSkewedtotherightandalsorangesbetween0and becomeTheannualmarginalprobabilityofdefaultofabondis15%inyear1and20%inyear2.WhatistheprobabilityoftheTheannualmarginalprobabilityofdefaultofabondis15%inyear1and20%inyear2.Whatistheprobabilityofthebondsurviving(i.e.nodefault)totheendoftwoyears?CorrectAnswer：Probability(nodefault)=(1–15%)×(1–20%)=OnamultiplechoiceexamwithfourchoicesforeachofOnamultiplechoiceexamwithfourchoicesforeachofsixquestions,whatistheprobabilitythatastudentgetslessthantwoquestionscorrectsimplybyguessing?p(X=0)=(3/4)6=p(X=1)=6×(1/4)×(3/4)5=Theprobabilityofgettinglessthantwoquestionscorrectis+p(X=1)=Acallcenterreceivesanaverageoftwophonecallsperhour.Theprobabilitythattheywillreceive20callsinanAcallcenterreceivesanaverageoftwophonecallsperhour.Theprobabilitythattheywillreceive20callsinan8-hourdayisclosestto:CorrectAnswer：Tosolvethisquestion,wefirstneedtorealizethattheexpectednumberofphonecallsinan8-hourdayis16.UsingthePoissondistribution,wesolvefortheprobabilitythatXwillbe20.=Ifwesaythatcommoditypricefollowalognormaldistribution,wemeanthatovertime:Ifwesaythatcommoditypricefollowalognormaldistribution,wemeanthatovertime:Thenaturallogarithmofthepriceisnormallydistributed.Thechangeinthepriceisnormallydistributed.Thechangeinthenaturallogarithmofthepriceisnormallydistributedovertime.ThereciprocalofthepriceisnormallyCorrectAnswer：Arandomvariablehasalognormaldistributionifitslogarithmisitselfnormallydistributed.andConfidenceandConfidenceSamplingandIntervalEstimateBestLinearUnbiasedEstimatorHypothesisThebasisofSamplingandIntervalEstimateBestLinearUnbiasedEstimatorHypothesisThebasisofTheapplicationof????SampleandSamplingandDescriptivestatistics:Summarizetheimportantcharacteristicsoflargedatasets. characteristicsofasmallerset(aSampleandSamplingandDescriptivestatistics:Summarizetheimportantcharacteristicsoflargedatasets. characteristicsofasmallerset(asamplepopulationStatisticalInference:EstimationandHypothesisSamplingandSimplerandomStratifiedrandomsampling:separate characteristics.StatisticalInference:EstimationandHypothesisSamplingandSimplerandomStratifiedrandomsampling:separate characteristics.Stratumandcells=M×Samplingsamplingerrorofthemean=samplemean–populationaaTheCentralLimitTheCentralLimitTheoremX1,X2,L,arandomsamplefromanypopulation2probabilitydistribution)withTheCentralLimitTheCentralLimitTheoremX1,X2,L,arandomsamplefromanypopulation2probabilitydistribution)with x,thesamplemeanσμXtendstobenormallydistributedwith/n2μσXxasthesamplesizeincreasesindefinitely(technically,infinitely)(≥Ofcourse,iftheXihappentobefromthenormalpopulation,thesamplemeanfollowsthenormaldistributionregardlessofthesampleStandardError(SE)ofmeanXHowever,thepopulation’sstandarddeviationisalmostneverknown.Instead,weusethestandarddeviationofthesampleSEX=nBestLinearUnbiasedEstimatorPropertiesofpointThemeanoftheestimatorscoincideswiththetrueparameterE(X)=Anunbiasedestimatorisalsoefficientifthevarianceofitssamplingdistributionissmallerthanalltheotherunbiasedestimatorsoftheparameteryouaretryingtoestimate.e.g.X~N(mXTheBestLinearUnbiasedEstimatorPropertiesofpointThemeanoftheestimatorscoincideswiththetrueparameterE(X)=Anunbiasedestimatorisalsoefficientifthevarianceofitssamplingdistributionissmallerthanalltheotherunbiasedestimatorsoftheparameteryouaretryingtoestimate.e.g.X~N(mXTheaccuracyoftheparameterestimateincreasesasthesamplesizeincreases(seethestandarderror).X==nn+BestLinearUnbiasedEstimatorAnotherpropertyofapointestimateislinearity.ApointestimateshouldbealinearestimatorBestLinearUnbiasedEstimatorAnotherpropertyofapointestimateislinearity.Apointestimateshouldbealinearestimator(i.e.,itcanbeusedasalinearfunctionofthesampledata).IftheestimatoristhebestavailablebestlinearunbiasedPointEstimationandConfidenceIntervalPointUsingasinglenumericalvaluetoestimatetheparameterofPointEstimationandConfidenceIntervalPointUsingasinglenumericalvaluetoestimatetheparameterofConfidenceIntervalLevelofsignificanceDegreeofconfidence(1–-0Thetdistributionfor27ConfidenceInterval=[PointEstimate+/-(reliabilityfactor)standardX S2fiXConfidenceIntervalThepopulationhasanormaldistributionwithaknownConfidencePointReliabilityStandardThepopulationhasanormaldistributionConfidenceIntervalThepopul