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(1<X<2)(b)Find:(1)E(X);,(2)GiventhatE(X2)=6.FindVar(X)10．Ifthejointprobabilitydensityfunctionoftworandomvariablesisgivenby(a)FindthemarginalpdfofXandY.(b)Determinewhetherthetworandomvariablesaredependentorindependent.11．SupposetherandomvariableXhasthedensityfunctionFindtheprobabilitydensityfunctionoftherandomvariableY=X+4.12.Takeoutanumberfrom1-200(200naturalnumbers).Find(a)theprobabilitythatthisnumbercanbedrivedby6;(b)theprobabilitythatthisnumbercanbedrivedby6aswellas8.13.Threechildrenareselectedatrandomfromagroupoffiveboysandthreegirls.(a)Whatistheprobabilitythatallthreeareboys?(b)Whatistheprobabilityatleasttwogirlsareselected?14.(a)Let(b)Let，，.Find.15.(a)SupposeA,Baretwoevents,andP(A)=1/4,P(B)=1/2,P(AB)=1/9,thenevaluate(b)UseBayes’TheoremtoshowthatifP(A),P(B)>0andthen16.(a)LetXhasaBinomialdistributionwithparametersnandp,i.e.X~b(n,p).Find(b)Showthat17.SupposethedistributionfunctionofarandomvariableXis，F(xiàn)ind(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<X<1).20.(a)Let.FindP(AB)(b)SupposeXandYareindependentrandomvariables,andFind21.Supposethedensityfunctionof(X,Y)is(a)Determinetheconstantk.(b)FindtheprobabilityP{X<1,Y<3}.22.LetXdenotethenumberoftimesacertainnumericalcontrolmachinewillmalfunction:1,2,or3timesonanygivenday.LetYdenotethenumberoftimesatechnicianiscalledonanemergencycall.TheirjointprobabilitydistributionisgivenasTable1.1Table1.1XY12310.050.050.120.050.10.35300.20.1(a)EvaluatethemarginaldistributionofXandY(b)DeterminewhetherthetworandomvariablesofXandYaredependentorindependent.23.(a)If,thenfind(b)Let(X,Y)be2-dimensionalrandomvariables,andIf，find.24.Supposetheprobabilitydistributionofisasfollows,and.FindthecorrelationcoefficientofXandY.25.LetXhasabeuniformdistributionontheinterval(0,1),andFind(a)(b)E(Y)26.TherandomvariableX,forfixed0<p<1andn≥1,hasprobabilitymassfunction(pmf)(a)Findcintermsofpandn.(b)FindthecumulativedistributionfunctionofX.27.If，,XandYareindependent.Find.28.Thelengthoftime,inminutes,foranairplanetoobtainclearancefortakeoffatacertainairportisarandomvariableY=3X-2,whereXhasthedensityfunctionFindthemeanandvarianceoftherandomvariableY.29.SupposetheprobabilitydensityfunctionofrandomvariableXis,and.Find(a)theprobabilitydensityfunctionofY,.(b)expectationE(Y)andvarianceVar(Y).30.SupposeXandYaretworandomvariables.Thejointpr