ProbabilityandStatisticsSolutions7GuanghuaSchoolofManagementPekingUniversityDecember7,20121/13Question1Forx∈[0,1],themarginaldistributionofXisf1(x)=�10(x+y)dy=x+12.ThentheconditionaldistributionofygivenX=xisf(y|x)=f(x,y)f1(x)=x+yx+12,0≤y≤1Thenforx∈[0,1],wehaveE(Y|X=x)=�∞−∞yf(y|x)dy=�10(x+y)yx+12dy=12x+13x+12.2/13Question1E(Y2|X=x)=�∞−∞y2f(y|x)dy=�10(x+y)y2x+12dy=13x+14x+12.ThenVar(Y|X=x)=E(Y2|X=x)−E2(Y|X=x)=112x2+112x+172(x+12)2.FromthefactthatVar{E(Y|X)}=E{E2(Y|X)}−(EY)2,wehaveE{E2(Y|X)}=�10E2(Y|X)f1(x)dx=�10(12x+13)2(x+12)2(x+12)dx=13+1144ln3.3/13Question1E(Y)=�10y(y+12)dy=712.ThenVar{E(Y|X)}=1144ln3−1144.BecauseE{Var(Y|X)}=Var(Y)−Var[E(Y|X)],E(Y2)=�10y2(y+12)dy=512.ThenVar(Y)=11144,E{Var(Y|X)}=112−ln3144.4/13Question2LetX1,X2denotethescoresoftwostudentsfromUniversityAandY1,Y2,Y3denotethescoresofthreestudentsfromUniversityB,theirdistributionsareX1,X2∼N(625,100)Y1,Y2,Y3∼N(600,150)So¯X∼N(625,50)¯Y∼N(600,50)Then¯X−¯Y∼N(25,100).5/13Question2SotheprobabilitythattheaverageofthescoresofthetwostudentsfromuniversityAwillbegreaterthantheaverageofthescoresofthethreestudentsformuniversityBPr(¯X−¯Y>0)=1−Pr(¯X−¯Y≤0)=1−Φ(0−2510)=1−Φ(−2.5)=Φ(2.5)=0.99386/13Question3SupposethemarginalandconditionaldistributionsareX∼N(µ0,σ20)Y|X=x∼N(ax+b,σ2).Thenf(x,y)=f(x)f(y|x)=1√2πσ0e−(x−µ0)22σ201√2πσe−(y−ax−b)22σ2=12πσ0σe−{(a22σ2+12σ20)x2+12σ2y2−aσ2xy+ex+gy+h}.7/13Frompage(233)exercise13,wehaveifthejointp.d.foftworandomvariablesXandYisproportionalasafunctionof(x,y)toexp(−[ax2+by2+cxy+ex+gy+h]),wherea>0,b>0,andab>(c/2)2.ThenXandYhaveabivariatenormaldistribution.Itiseasytosee(a22σ2+12σ20)12σ2>(a2σ2)2=a24σ4ThenX,Yhaveabivariatenormaldistribution.8/13Question4(1)Thelog-likelihoodfunctionisℓ(θ)=n�i=1ln{f(xi;θ)}=n�i=1ln{θxie−θxi!}=n�i=1{xilnθ−θ+ln(xi!)}xi=0,1,2,...Thenlet∂lnℓ(θ)∂θ=n�i=1(xiθ−1)=0Sowehaveˆθ=¯x.Itiseasytocheck∂2lnℓ∂θ2|θ=ˆθ=−n�i=1xiθ2|θ=ˆθ≤0.9/13Question4SoˆθistheM...
