Homework61.SupposethatXandYhaveacontinuousjointdistributionforwhichthejointp.d.f.isasfollows:f(x,y)=�������12y2,for0≤y≤x≤10,otherwise.Determineρ(X,Y)(thecorrelationofXandY).2.SupposethatX,Y,andZarethreerandomvariablessuchthatVar(X)=1,Var(Y)=4,Var(Z)=8,Cov(X,Y)=1,Cov(X,Z)=−1andCov(Y,Z)=2.Determine(a)Var(X+Y+Z)and(b)Var(3X−Y−2Z+1)3.(TextbookSection4.7-6,Page178)SupposethatX1,···,Xnfromarandomsampleofsizenfromadistributionforwhichthemeanis6.5andthevarianceis4.Determinehowlargethevalueofnmustbeinorderforthefollowingrelationtobesatisfied.Pr(6≤Xn≤7)≥0.8.4.LetZ1,Z2,...beasequenceofrandomvariables;andsupposethat,forn=1,2,···1thedistributionofZnisasfollows:Pr(Zn=n2)=1nandPr(Zn=0)=1−1nShowthatlimn→∞E(Zn)=∞butZnp−−−→05.SupposethatXhasanormaldistributionforwhichthemeanis1andthevarianceis4.Findthevalueofeachofthefollowingprobabilities.1)Pr(X≤3)2Pr(X=1)3)Pr(−1
