概率论与数理统计英文题目

Test1.ConsiderthesetwithsubsetsFindthefollowingsets:(a)and(b)and2.ShowDeMorgansLaw:(a),(b)3.(a)LetP(A)=0.5,P(B)=0.4,P(A∩B)=0.2.FindP(A∪B)andP(A|B)(b)ConsidertwofairdiceAandB.DieAissix-sidedandisnumbered1throughto6whilstdieBisfour-sidedandisnumbered1through4.Bothdicearerolled.Findtheprobabilityoftwodiceshowthesamescore.4.(a).LetXbearandomvariable..Showthat(b)ThepmfofarandomvariableXwhichhasaPoissondistributionwithparameter2.FindP(X=3).5.Suppose(X,Y)bebivariatenormaldistribution..FindthecorrelationcoefficientofXandY.6.(a)LetXUniformlydistributedon(-1,1),i.e.X~U(-1,1).FindtheExpectationandVarianceofX.(b)LetXandYbecontinuousrandomvariableswithjointpdfLetZ=X+Y,Find.Inparticular,ifXandYareindependent,FindfX*fY..7.(a)TherandomvariableXhaspmfis:X0123PX(x)0.20.160.410.23FindP(1≤X<3)andFX(1.7).(b)LetXbeacontinuousrandomvariablewithpdfFindP(X=3),P(X>1)and.8.Xiscontinuousr.v.withpdfLet,FindE(Y)9.LetXbeacontinuousrandomvariablewithprobabilitydensityfunction(a)FindP(10andthen16.(a)LetXhasaBinomialdistributionwithparametersnandp,i.e.X~b(n,p).Find(b)Showthat17.SupposethedistributionfunctionofarandomvariableXis，Find(a)theprobabilitythatXgetsvaluewithin(0.3，0.7);(b)thedensityfunctionofX18.TheoperationallifetimeX,inyears,ofabatterypoweredwatchhasprobabilitydensityfunction(a)Findthevalueofc.(b)FindthecumulativedistributionfunctionofX.(c)Findtheprobabilitythatthewatchhasanoperationallifetimeinexcessof4years.19.(a)TherandomvariableXhastheprobabilitymassfunctionbelow:X1234Px(x)0.40.14a0.24Findaand(b)AcontinuousrandomvariableXhavingtheprobabilitydensityfunction.FindandP(0