Mathematical Statistics with Applications, 7 edition ISM_Chapter06F

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1、Chapter6:FunctionsofRandomVariablesy26.1ThedistributionfunctionofYisF(y)=2(1−t)dt=2y−y,0≤y≤1.Y∫0a.F(u)=P(U≤u)=P(2Y−1≤u)=P(Y≤u+1)=F(u+1)=2(u+1)−(u+1)2.Thus,U112Y222f(u)=F′(u)=1−u,−1≤u≤1.U1U12b.F(u)=P(U≤u)=P(1−2Y≤u)=P(Y≤1−u)=F(1−u1)=1−2(u+1)=(u+1)2.Thus,U222Y222f(u)=F′(u)=u+1,−1≤u≤1.U2U222c.F(u)

2、=P(U≤u)=P(Y≤u)=P(Y≤u)=F(u)=2u−uThus,U33Yf(u)=F′(u)=1−1,0≤u≤1.U3U3ud.E(U)=−1/3,E(U)=1/3,E(U)=1/6.1232e.E(2Y−1)=−1/3,E(1−2Y)=1/3,E(Y)=1/6.y236.2ThedistributionfunctionofYisF(y)=(3/2)tdt=(1/2)(y−1),–1≤y≤1.Y∫−1a.()()(3)(/3)(/3)1(3/181).Thus,Fu=PU≤u=PY≤u=PY≤u=Fu=u−U11Y22f(u)=F′(u)=u/18,−3≤u≤3.U1U11

3、3b.F(u)=P(U≤u)=P(3−Y≤u)=P(Y≥3−u)=1−F(3−u)=[1−(3−u)].U22Y232Thus,f(u)=F′(u)=(3−u),2≤u≤4.U2U2223/2c.F(u)=P(U≤u)=P(Y≤u)=P(−u≤Y≤u)=F(u)−F(−u)=u.U33YYThus,f(u)=F′(u)=3u,0≤u≤1.U3U322⎧y/20≤y≤1⎪6.3ThedistributionfunctionforYisFY(y)=⎨y−1/211.5⎩a.F(u)=P(U≤u)=P(10Y−4≤u)=P(Y≤u+4)=F(u+4).So,U10Y

4、102⎧(u+4)−4≤u≤6⎧u+4−4≤u≤6200100⎪⎪F(u)=u−1611⎪⎩0elsewhere⎩b.E(U)=5.583.c.E(10Y–4)=10(23/24)–4=5.583.−y/46.4ThedistributionfunctionofYisF(y)=1−e,0≤y.Ya.F(u)=P(U≤u)=P(3Y+1≤u)=P(Y≤u−1)=F(u−1)=1−e−(u−1)/12.Thus,U3Y31−(u−1)/12f(u)=F′(u)=e,u≥1.UU12b.E(U)=13.12

5、1122Chapter6:FunctionsofRandomVariablesInstructor’sSolutionsManual6.5ThedistributionfunctionofYisF(y)=y/4,1≤y≤5.YF(u)=P(U≤u)=P(2Y2+3≤u)=P(Y≤u−3)=F(u−3)=1u−3.Differentiating,U2Y242f(u)=F′(u)=1()u−3−1/2,5≤u≤53.UU1626.6RefertoEx.5.10ad5.78.DefineF(u)=P(U≤u)=P(Y−Y≤u)=P(Y≤Y+u).U1212a.Foru≤0,F(u)=P(

6、U≤u)=P(Y−Y≤u)=0.U12uy2+u2For0≤u<1,F(u)=P(U≤u)=P(Y−Y≤u)=1dydy=u/2.U12∫∫1202y22−u22For1≤u≤2,F(u)=P(U≤u)=P(Y−Y≤u)=1−1dydy=1−(2−u)/2.U12∫∫120y2+u⎧u0≤u<1⎪Thus,fU(u)=FU′(u)=⎨2−u1≤y≤2.⎪⎩0elsewhereb.E(U)=1.6.7LetFZ(z)andfZ(z)denotethestandardnormaldistributionanddensityfunctionsrespectively.2a.F(u)=P(

7、U≤u)=P(Z≤u)=P(−u≤Z≤u)=F(u)−F(−u).TheUZZdensityfunctionforUisthenf(u)=F′(u)=1f(u)+1f(−u)=1f(u),u≥0.UU2uZ2uZuZ1−1/2−u/2Evaluating,wefindf(u)=ueu≥0.Uπ2b.Uhasagammadistributionwithα=1/2andβ=2(recallthatΓ(1/2)=π).c.Thisisthechi–squaredistrib