2026年CFA二级《数量方法》真题及答案

2026年CFA二级《数量方法》真题及答案

一、单项选择题（总共10题，每题2分）1.Whichofthefollowingisleastlikelyapropertyofaprobabilitydistribution?A.Thesumofallprobabilitiesmustequal1.B.Theprobabilityofaneventmustbebetween0and1inclusive.C.Theprobabilityofaneventcanbenegative.2.ArandomvariableXfollowsanormaldistributionwithameanof10andastandarddeviationof2.WhatistheprobabilitythatXisgreaterthan12?A.0.1587B.0.3413C.0.65873.ThecovariancebetweentworandomvariablesXandYis5.ThestandarddeviationofXis2andthestandarddeviationofYis3.WhatisthecorrelationcoefficientbetweenXandY?A.0.8333B.0.5556C.0.33334.Whichofthefollowingmeasuresofcentraltendencyismostaffectedbyextremevalues?A.MeanB.MedianC.Mode5.Asampleofsizen=25isdrawnfromapopulationwithameanof50andastandarddeviationof10.Whatisthestandarderrorofthesamplemean?A.0.4B.2C.46.Thenullhypothesisisthatthepopulationmeanisequalto100.Thealternativehypothesisisthatthepopulationmeanisgreaterthan100.Asampleofsizen=36isdrawnandthesamplemeanis105.Thepopulationstandarddeviationisknowntobe12.Whatisthevalueoftheteststatistic?A.2.5B.1.25C.0.41677.Aconfidenceintervalforthepopulationmeanisconstructedwitha95%confidencelevel.Thesamplemeanis50andthemarginoferroris5.Whatisthelowerlimitoftheconfidenceinterval?A.45B.50C.558.Whichofthefollowingisanon-parametrictest?A.t-testB.ANOVAC.Wilcoxonsigned-ranktest9.Thechi-squaretestofindependenceisusedtodetermineifthereisarelationshipbetweentwocategoricalvariables.Thedegreesoffreedomforachi-squaretestofindependencewith3rowsand4columnsis:A.6B.7C.810.Thecorrelationcoefficientbetweentwovariablesis0.8.Whatpercentageofthevariationinonevariableisexplainedbythevariationintheothervariable?A.80%B.64%C.40%二、填空题（总共10题，每题2分）1.TheprobabilityofaneventAgiventhateventBhasoccurrediscalledthe______probability.2.ArandomvariableXfollowsauniformdistributionovertheinterval[a,b].TheprobabilitydensityfunctionofXisgivenbyf(x)=______.3.TheexpectedvalueofadiscreterandomvariableXisgivenbyE(X)=______.4.ThestandarddeviationofadiscreterandomvariableXisgivenbyσ(X)=______.5.ThecovariancebetweentwoindependentrandomvariablesXandYis______.6.Thecoefficientofdetermination,R²,isameasureofthe______oftheregressionline.7.TheF-testisusedtotestthesignificanceofthe______inaregressionmodel.8.Thedegreesoffreedomforat-testwithasamplesizeofn=20andone-tailedalternativehypothesisis______.9.Thecriticalvalueforachi-squaretestwith10degreesoffreedomandasignificancelevelof0.05is______.10.Theconfidenceintervalforthepopulationproportionisgivenbyp̂±______.三、判断题（总共10题，每题2分）1.Thesumoftheprobabilitiesofallpossibleoutcomesofarandomexperimentisalwaysequalto1.2.Themeanofabinomialdistributionisnp,wherenisthenumberoftrialsandpistheprobabilityofsuccessineachtrial.3.Themedianofaskeweddistributionisalwaysclosertothemeanthanthemode.4.Thestandarderrorofthesamplemeandecreasesasthesamplesizeincreases.5.ATypeIerroroccurswhenwerejectthenullhypothesiswhenitisactuallytrue.6.Aconfidenceintervalforthepopulationmeanisalwayssymmetricaroundthesamplemean.7.Thet-distributionissymmetricandhasameanof0.8.TheF-distributionisalwayspositive.9.Thechi-squaretestofgoodness-of-fitisusedtotestifasamplecomesfromaparticulardistribution.10.Thecorrelationcoefficientmeasuresthestrengthanddirectionofthelinearrelationshipbetweentwovariables.四、简答题（总共4题，每题5分）1.Explainthedifferencebetweenadiscreterandomvariableandacontinuousrandomvariable.2.Describethestepsinvolvedinhypothesistesting.3.Whatisthedifferencebetweenaone-tailedtestandatwo-tailedtest?4.Explaintheconceptofmulticollinearityinregressionanalysis.五、讨论题（总共4题，每题5分）1.Discusstheimportanceofprobabilityinfinancialdecision-making.2.Howcanweuseprobabilitydistributionstomodelinvestmentreturns?3.Explaintheroleofhypothesistestinginfinancialresearch.4.Discussthelimitationsofusingregressionanalysisinfinancialforecasting.答案及解析一、单项选择题1.C.Theprobabilityofaneventmustbebetween0and1inclusive.Aprobabilitycannotbenegative.2.A.First,westandardizethevalue12usingtheformulaz=(x-μ)/σ,wherex=12,μ=10,andσ=2.z=(12-10)/2=1.Lookinguptheprobabilityinthestandardnormaltable,P(Z>1)=0.1587.3.A.Thecorrelationcoefficientisgivenbyρ=cov(X,Y)/(σ(X)σ(Y)).Substitutingthegivenvalues,ρ=5/(23)=0.8333.4.A.Themeanismostaffectedbyextremevaluesasittakesintoaccountallthedatapoints.5.A.Thestandarderrorofthesamplemeanisgivenbyσ/√n,whereσisthepopulationstandarddeviationandnisthesamplesize.Substitutingthegivenvalues,σ/√n=10/√25=2.6.A.Theteststatisticforaone-samplez-testisz=(x̄-μ)/(σ/√n),wherex̄=105,μ=100,σ=12,andn=36.z=(105-100)/(12/√36)=2.5.7.A.Thelowerlimitoftheconfidenceintervalisgivenbyx̄-E,wherex̄=50andE=5.So,thelowerlimitis50-5=45.8.C.TheWilcoxonsigned-ranktestisanon-parametrictest.Thet-testandANOVAareparametrictests.9.A.Thedegreesoffreedomforachi-squaretestofindependencewithrrowsandccolumnsis(r-1)(c-1).Here,r=3andc=4,so(3-1)(4-1)=6.10.B.Thecoefficientofdetermination,R²,isthesquareofthecorrelationcoefficient.So,ifthecorrelationcoefficientis0.8,R²=0.64,whichmeans64%ofthevariationinonevariableisexplainedbythevariationintheothervariable.二、填空题1.Conditional2.1/(b-a)fora≤x≤band0otherwise3.Σ(xP(x)),wherethesumistakenoverallpossiblevaluesofx4.√[Σ((x-E(X))²P(x))],wherethesumistakenoverallpossiblevaluesofx5.06.Goodness-of-fit7.Overallsignificanceoftheregression8.199.18.30710.zα/2√[p̂(1-p̂)/n]三、判断题1.True2.True3.False.Inaskeweddistribution,themeanisusuallyfartherfromthemedianthanthemode.4.True5.True6.True7.True8.True9.True10.True四、简答题1.Adiscreterandomvariablecantakeononlyacountablenumberofdistinctvalues,suchasthenumberofheadsinaseriesofcoinflips.Acontinuousrandomvariablecantakeonanyvaluewithinacertaininterval,suchastheheightofaperson.Theprobabilitydistributionofadiscreterandomvariableisgivenbyaprobabilitymassfunction,whiletheprobabilitydistributionofacontinuousrandomvariableisgivenbyaprobabilitydensityfunction.2.Thestepsinhypothesistestingare:-Statethenullhypothesis(H₀)andthealternativehypothesis(H₁).-Selectthesignificancelevel(α).-Determinetheappropriateteststatistic.-Calculatethevalueoftheteststatistic.-Determinethecriticalvalueorp-value.-Comparetheteststatisticwiththecriticalvalueorp-valuewithα.-Makeadecisiontorejectorfailtorejectthenullhypothesis.3.Aone-tailedtestisusedwhenweareinterestedintestingwhetheraparameterisgreaterthanorlessthanacertainvalue.Atwo-tailedtestisusedwhenweareinterestedintestingwhetheraparameterisdifferentfromacertainvalue(eithergreaterorless).Inaone-tailedtest,thecriticalregionisinonetailofthedistribution,whileinatwo-tailedtest,thecriticalregionissplitbetweenthetwotails.4.Multicollinearityinregressionanalysisoccurswhentwoormoreindependentvariablesarehighlycorrelatedwitheachother.Thiscancauseproblemssuchasunstableestimatesoftheregressioncoefficients,difficultyininterpretingtheindividualeffectsoftheindependentvariables,andincreasedvarianceoftheestimates.五、讨论题1.Probabilityisimportantinfinancialdecision-makingasithelpsinquantifyinguncertainty.Itallowsinvestorstoassessthelikelihoodofdifferentoutcomesandmakemoreinformeddecisions.Forexample,probabilitycanbeusedtoestimatetheprobabilityofastockpriceincreaseordecrease,theprobabilityofdefaultofaborrower,ortheprobabilityofaparticulareconomicevent.2.Wecanuseprobabilitydistributionstomodelinvestmentreturns.Forexample,thenormaldistributionisoftenusedtomodelstockreturns.Byknowingthemeanandstandarddeviationofthereturns,wecanestimatetheprobabilityofdifferentreturnlevels.Otherdistributionssuchasthelognormaldistributioncanalsobeusedtomodelassetprices.3.Hypothesistestingisusedinfinancialresearchtotestvarioustheoriesandhypotheses.Forexample,wecantestwhethera