2021CFA二级《数量方法》真题及答案无错误校正版

2021CFA二级《数量方法》真题及答案无错误校正版

一、单项选择题（总共10题，每题2分）1.Asampleof50observationshasameanof10andastandarddeviationof2.Whatisthestandarderrorofthemean?A.0.2828B.0.4C.22.TheprobabilityofaneventAis0.4,theprobabilityofeventBis0.3,andtheprobabilityofAandBoccurringtogetheris0.1.WhatistheprobabilityofAorBoccurring?A.0.6B.0.7C.0.83.Whichofthefollowingisacharacteristicofanormaldistribution?A.Itisskewedtotheright.B.Themean,median,andmodeareequal.C.Ithasheavytails.4.Aportfoliohasanexpectedreturnof12%andastandarddeviationof15%.Therisk-freerateis3%.TheSharperatiooftheportfoliois:A.0.6B.0.8C.1.05.IfthecorrelationcoefficientbetweentwovariablesXandYis0.8,andthestandarddeviationofXis0.5andthestandarddeviationofYis0.6,whatisthecovariancebetweenXandY?A.0.24B.0.48C.0.326.A95%confidenceintervalforthepopulationmeanisconstructedwhenthepopulationstandarddeviationisknown.Ifthesamplesizeisincreasedfrom100to400,thewidthoftheconfidenceintervalwill:A.IncreaseB.DecreaseC.Remainthesame7.TheexpectedvalueofadiscreterandomvariableXwithprobabilities\(P(X=1)=0.3\),\(P(X=2)=0.4\),\(P(X=3)=0.3\)is:A.1.5B.2.0C.2.58.Acompany'ssalesinthelast5yearsare\(100,120,130,140,150\).Whatisthecompoundannualgrowthrate(CAGR)?A.10.0%B.11.4%C.12.0%9.Whichofthefollowingisapropertyofthechi-squaredistribution?A.Itissymmetric.B.Themeanisequaltothedegreesoffreedom.C.Itispositivelyskewed.10.Inasimplelinearregressionmodel\(y=\beta_0+\beta_1x+\epsilon\),iftheslopecoefficient\(\beta_1\)is2,andanewobservation\(x=5\)isadded,whatistheexpectedchangein\(y\)foraone-unitincreasein\(x\)?A.2B.5C.10二、填空题（总共10题，每题2分）1.Theformulaforthesamplevarianceis\(\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}\),where\(x_i\)arethe__________and\(\bar{x}\)isthe__________.2.Theprobabilityofaneventnotoccurringis\(1-P(A)\),where\(P(A)\)isthe__________ofeventA.3.Forabinomialdistributionwith\(n=10\)and\(p=0.2\),themeaniscalculatedas__________.4.Thestandardnormaldistributionhasameanof__________andastandarddeviationof__________.5.Thecoefficientofvariationiscalculatedas\(\frac{\text{__________}}{\text{Mean}}\times100\%\).6.Inanormaldistribution,theareaunderthecurvebetween\(\mu-\sigma\)and\(\mu+\sigma\)isapproximately__________.7.Theformulaforthepresentvalueofanordinaryannuityis\(PV=A\times\frac{1-(1+r)^{-n}}{r}\),where\(A\)isthe__________,\(r\)isthe__________,and\(n\)isthe__________.8.Ifthecorrelationcoefficientbetweentwovariablesis0,itimpliesthatthereis__________linearrelationshipbetweenthem.9.Theformulaforthedegreesoffreedominaone-sample\(t\)-testis__________.10.Inasimplelinearregression,thesumofsquaredresidualsisdenotedas__________.三、判断题（总共10题，每题2分）1.Asamplingdistributionisthedistributionofasamplestatistic.2.Theprobabilityofanimpossibleeventis1.3.Auniformdistributionhasaconstantprobabilitydensityfunction.4.AhigherSharperatioindicatesamoreriskyportfolio.5.Thecoefficientofdetermination\(R^2\)rangesfrom0to1.6.At-distributionhasheaviertailsthananormaldistribution.7.Inanormaldistribution,themedianisalwaysgreaterthanthemean.8.Thesamplevarianceisanunbiasedestimatorofthepopulationvariance.9.Iftwoeventsaremutuallyexclusive,then\(P(A\capB)=P(A)+P(B)\).10.Thestandarderroroftheestimateinregressionmeasurestheaccuracyoftheregressionlineinpredictingthedependentvariable.四、简答题（总共4题，每题5分）1.Explaintheconceptofsamplingerror.Howcanitbereduced?2.Differentiatebetweenadiscreteandacontinuousrandomvariable.Giveanexampleofeach.3.Describethepropertiesofthenormaldistributionanditsimportanceinfinance.4.Whatisthedifferencebetweenasimplelinearregressionandamultiplelinearregression?五、讨论题（总共4题，每题5分）1.Discussthefactorsthataffectthechoiceofconfidencelevelinconstructingaconfidenceinterval.2.Analyzetheimpactofcorrelationbetweenindependentvariablesonamultiplelinearregressionmodel.3.Explaintheconceptoftheefficientfrontierinportfoliotheory.Howisitrelatedtorisk-returntrade-off?4.Considerascenariowhereacompanyisconsideringtwoinvestmentprojects.Discussthecriteriathatcanbeusedtochoosebetweenthem,takingintoaccountriskandreturn.答案单项选择题1.A(Standarderrorofthemean\(=\frac{\sigma}{\sqrt{n}}=\frac{2}{\sqrt{50}}\approx0.2828\))2.A(\(P(A\cupB)=P(A)+P(B)-P(A\capB)=0.4+0.3-0.1=0.6\))3.B(Inanormaldistribution,themean,median,andmodeareequal)4.A(Sharperatio\(=\frac{R_p-R_f}{\sigma_p}=\frac{12\%-3\%}{15\%}=0.6\))5.B(Covariance\(\rho_{XY}=\text{Corr}(X,Y)\times\sigma_X\times\sigma_Y=0.8\times0.5\times0.6=0.48\))6.B(Asthesamplesize\(n\)increases,thestandarderrorofthemean\(\frac{\sigma}{\sqrt{n}}\)decreases,sothewidthoftheconfidenceintervaldecreases)7.B(\(E(X)=1\times0.3+2\times0.4+3\times0.3=2.0\))8.B(UsingtheCAGRformula\(\text{CAGR}=\left(\frac{150}{100}\right)^{\frac{1}{4}}-1\approx0.114=11.4\%\))9.C(Thechi-squaredistributionispositivelyskewed)10.A(Intheregressionmodel\(y=\beta_0+\beta_1x+\epsilon\),theslope\(\beta_1\)representsthechangein\(y\)foraone-unitchangein\(x\))填空题1.sampleobservations;samplemean2.probability3.\(n\timesp=10\times0.2=2\)4.0;15.standarddeviation6.68%7.annuitypayment;interestrate;numberofperiods8.no9.\(n-1\)10.\(SSE\)判断题1.True2.False(probabilityofanimpossibleeventis0)3.True4.False(ahigherSharperatioindicatesamorefavorablerisk-returntrade-off)5.True6.True7.False(inanormaldistribution,themean,median,andmodeareequal)8.True9.False(\(P(A\capB)=0\)formutuallyexclusiveevents,\(P(A\cupB)=P(A)+P(B)\))10.True简答题1.Samplingerroristhedifferencebetweenthesamplestatisticandthecorrespondingpopulationparameter.Itcanbereducedbyincreasingthesamplesize,usingarandomandrepresentativesample.2.Adiscreterandomvariablecanonlytakeonacountablenumberofvalues,likethenumberofchildreninafamily.Acontinuousrandomvariablecantakeonanyvaluewithinaninterval,liketheheightofaperson.3.Thenormaldistributionissymmetric,bell-shapedwithmean=median=mode.Itisimportantinfinanceformodelingreturns,asmanyfinancialvariablesareassumedtofollowanormaldistribution.4.Simplelinearregressionhasoneindependentvariable,whilemultiplelinearregressionhastwoormoreindependentvariables.讨论题1.Factorsincludethelevelofcon