爱考机构北大考研辅导班：北大光华统计讲义

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1、Exercises1.AboutMLE1-1.Supposethatformarandomsamplefromadistributionforwhichthep.d.f.isasfollowsAlso,supposethatthevalueofisunknown(.a)ShowthattheMLEofdoesnotexist.b)Determineanotherversionofthep.d.f.ofthissamedistributionforwhichtheMLEofwillexist,andfindthisestimator

2、.1-2.Supposethatformarandomsamplefromadistributionforwhichthep.d.f.asfollows:Also,supposethatthevalueofisunknown(.FindtheMLEof1-1.Supposethatformarandomsamplefromadistributionforwhichthep.d.f.isasfollowsAlso,supposethatthevalueofisunknown.FindtheMLEof1-2.Supposethatth

3、elifetimeofacertaintypeoflamphasanexponentialdistributionforwhichthevalueoftheparameterisunknown.ArandomsampleofnlampsofthistypearetestedforaperiodofThoursandthenumberXoflampsthatfailduringthisperiodisobserved;butthetimesatwhichthefailuresoccurredarenotnoted.Determine

4、theMLEofbasedontheobservedvalueofX1-3.Supposethatformarandomsamplefromauniformdistributionontheinterval[a,b],wherebothendpointsaandbareunknown.FindtheMLEofthemeanofthedistribution.1-1.Supposethatformarandomsamplefromanormaldistributionforwhichboththemeanandthevariance

5、areunknown.FindtheMLEofthe0.95quantileofthedistribution.Thatis,ofthepointsuchthat1-2.Supposethatformarandomsampleofsizenfromauniformdistributionontheinterval,wherethevalueofisunknown.ShowthatthesequenceofMLEsofisaconsistentsequence.1.AboutEstimation2-1.Supposethataran

6、domsampleistobetakenfromaBernoullidistributionforwhichthevalueoftheparameterpisunknown.Supposealsothatitisbelievedthatthevalueofpisintheneighborhoodof0.2.Howlargearandomsamplemustbetakeninorderthatwhenp=0.2?Fortheconditionsintheaboveexercise,usethecentrallimittheoremt

7、ofindapproximatelythesizeofarandomsamplethatmustbetakeninorderthatwhenp=0.2?Fortheconditionsintheaboveexercise,howlargearandomsamplemustbetakeninorderthatwhenp=0.2?2-2.Supposethattherandomvariablesareindependent,andeachrandomvariablehasacontinuousd.f..Also,lettherando

8、mvariableYbedefinedbytherelation.ShowthatYhasadistributionwith2ndegreesoffreedom.2-3.Supposethatformarandomsamplefromanormal