Homework71.SupposethatXandYhaveacontinuousjointdistributionforwhichthejointp.d.f.isasfollows:f(x,y)=�������x+y,for0≤x≤1,and0≤y≤10,otherwise.(1)FindE(Y|X)andVar(Y|X).(1)FindVar[E(Y|X)]andE[Var(Y|X)].2.Supposethatonacertainexaminationinadvancedmathematics,studentsfromuniver-sityAachievescoresthatarenormallydistributedwithameanof625andavarianceof100,andstudentsfromuniversityBachievesscoreswhicharenormallydistributedwithameanof600andavarianceof150.IftwostudentsfromuniversityAandthreestudentsfromuniversityBtakethisexamination,whatistheprobabilitythattheaverageofthescoresofthetwostudentsfromuniversityAwillbegreaterthantheaverageofthescoresofthethreestudentsformuniversityB?Hint:Determinethedistributionofthedifferencebetweenthetwoaverages.13.SupposethatarandomvariableXhasanormaldistribution,andforeveryx,theconditionaldistributionofanotherrandomvariableYgivenX=xisanormaldistri-butionwithmeanax+bandvarianceσ2,wherea,bandσ2areconstants.ProvethatthejointdistributionofXandYisabivariatenormaldistribution.4.LetX1,X2,...,Xnrepresentarandomsamplefromeachofthedistributionshavingthefollowingprobabilitydensityfunctions:(a)f(x;θ)=θxe−θ/x!,x=0,1,2,...,0≤θ<∞,zeroelsewhere,wheref(0;0)=1.(b)f(x;θ)=θxθ−1,00andθ2>0IfX1,X2,...,Xnisarandomsamplefromthisdistribution,findthemaximumlikeli-hoodestimatorsofθ1andθ2.2
