应用统计学课件 33页

1.2.
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A(2)GH23(X1,···,Xn)B(x1,···,xn)A=H(3)=(X1,···,Xn)=?<;A=@)(SampleSpace),AX.B(x1,···,xn)∈X4.CI=DCD(1)J5X1,···,XnXi/4;XE)F?(2)FX1,···,XnEGFE?1X∼F(x)(X1,···,Xn)
4F:nnF(xi),
4I!:f(xi).i=1i=122MFJK GN$HFN(μ,σ),>LI!:1(x−μ)2f(x)=√exp{−},−∞.E)MN:ci−1+ci 2,4.Æx#O(ci,i=0,1,···,m,RN)(ci−1,ci]finiTQLOUyi==VRPhnhIAQL/I4\n351255?WS5QU0"25QL/IXXXY1.1RS$VZ[(%)1.401.281.361.381.441.401.341.541.441.461.80∗1.441.461.501.381.541.501.481.521.581.521.461.421.581.701.621.581.621.761.681.681.661.621.721.601.621.461.381.421.381.601.441.461.381.341.381.341.361.581.381.341.281.081.081.361.501.461.281.181.281.261.501.521.381.501.521.501.461.341.401.501.421.381.361.381.421.341.481.361.361.321.401.401.261.261.161.341.401.161.541.241.221.201.301.361.301.481.281.181.281.301.521.761.161.281.481.461.481.421.361.321.221.721.181.361.441.281.101.06∗1.101.161.221.241.221.34Y1.2RS[VXTY\]%^%$&XT]%^%$&XT0.99–1.091.0431.39–1.491.44291.09–1.191.1491.49–1.591.54191.19–1.291.24181.59–1.691.6491.29–1.391.34321.69–1.791.7451.79–1.891.841(1)QL/IYZ1,(2)ÆR7[hist`(3)I'"U8",QL/ `1nII9x¯=ni=1xi;5\nIII\a,+WX]^J)%Y?Z1(X1,···,Xn)_(;X=T=T(x1,···,xn)=@)XO[:&T(X1,···,Xn)UE\*`])AT(X1,···,Xn)(Statistics).24.X1,X2$;N(μ,σ)2=9μ_)σ)$CR`a%22222X1·X2−3σ,X1+X2+5σ,X1,X1+5X2,X1/X2+3μX1(1)b*=P+0abcd+c9ed$bcf(2)EZ2(X1,···,Xn)(;X,+n=1n(1)X=nΣi=1XiA=`(2)S2=1n(X−X)2n−1Σi=1iA=/)1n2bS=Σi=1(Xi−X)=/)<=(n−1)`6\nn1k(3)Ak=XiA=kgR`ni=1n1kBk=(Xi−X)=kgeO?ni=1S˜2,S˜2=1n2x},{νn(x)=n}={X(n)≤x},{νn(x)=k}={X(k)≤x2).⎧⎪⎪⎪⎪⎪⎪0,x≤0⎪⎪⎪⎪⎨2,f-F(x)=⎪⎪x,01b⎛⎞44⎜⎜⎟⎟k4−kF3(x)=⎝⎠[F(x)][1−F(x)]k=3k34=4[F(x)][1−F(x)]+[F(x)]34=4[F(x)]−3[F(x)]⎧⎪⎪⎪⎪⎪⎪0,x≤0⎪⎪⎪⎪⎨=4x6−3x8,01111P(X(3)>)=1−P(X(3)≤)=1−F3()222161813243=1−(4()−3())=1−=2225625612\n⎧⎪⎪⎪⎪⎪⎨6x(1−x),015!22f3(x)=[F(x)][1−F(x)]·f(x)2!2!22=30[F(x)][1−F(x)]·f(x)⎧⎪⎪⎪⎪⎪⎪0,x≤0⎪⎪⎪⎪⎨=180x(1−x)(3x2−2x3)2(1−3x2+2x3)2,01⎧⎪⎪⎪⎪52232⎪⎨180x(1−x)(3−2x)(1−3x+2x),00P/β>0k!(X1,···,Xn)(;=0qX(1)oN$WeibullF?0qX(1)F:⎡⎤nxαF1(x)=1−[1−F(x)]=1−exp⎣−n()⎦β⎡⎛⎞α⎤⎢⎜x⎟⎥=1−exp⎣−⎝1⎠⎦−nαβ−1αX(1)N$P/oα,k!nβWeibullF?(<)GX1.;XI!:f(x),(X1,···,Xn)(;=]pGXX(i)i)EGF?0q:(X(i),X(j))
4FI!:n!−xi−1−x−yj−i−1fij(x,y)=[1−eθ][eθ−eθ](i−1)!(j−i−1)!(n−j)!−yn−j1−x1−y·[eθ]eθeθ,00,v>0n!−ui−1−un−i1−ufU(u)=[1−eθ](eθ)eθ,u>0(i−1)!(n−i)!θ8X(i)FI!:b(n−i)!−vj−i−1−vn−j1−vfV(v)=[1−eθ](eθ)eθ,v>0(j−i−1)!(n−j)!θ16\n8V=X(j)−X(i)FI!:-bX(i)X(j)−X(i)(j>i)EGF?6;XFI!:⎧⎪⎪⎪⎪2⎪⎨3x,0≤x≤1f(x)=⎪⎪⎪⎪⎪⎩0,9iX(1)≤X(2)≤···≤X(n)(;=GX0qX(2)/X(4)X(4)EGFEY?=i)1.Z(X1,···,Xn)(;X=(X(1),···,X(n))GXARn=X(n)−X(1)=i)?;%F&!2.(X1,···,Xn)(;X=f(x),F(x)cXFI!:F:=i)RnFI!:F:c∞n−2fRn(x)=n(n−1)[F(x+u)−F(u)]f(x+u)f(u)du,x>0,−∞∞n−1FRn(x)=n[F(x+u)−F(u)]f(u)du,x>0.−∞3.Z&EXF:F(x),]17\n00,c(X(n)−an)/bngquirFG(x),AG(x)i,F.L=Zi+Fi,Fi+FAiF?I!i,F(UAopq(Gumbel)F)−xG1(x)=exp(−e),−∞0;G2(x)=⎪⎪⎪⎪⎪⎩0,x≤0,9k>0?III!i,F(UAsFq(Weibull)F)⎧⎪⎪⎪⎪⎪⎨1,x>0;G3(x)=⎪⎪⎪⎪k⎪⎩exp(−(−x)),x≤0,9k>0?sZiF−1/ξG(x;θ)=exp[−{1+ξ(x−μ)/σ}+]θ=(μ,σ,ξ)Dσ>0,μ,ξ∈R,\]ps,{s}+=max{s,0}.AGEV(μ,σ,ξ)Rt5Æevd,evir,ismev,evdbayes,extremes2/i?19\nhDF3F(x;θ)5?XFθ=?4A@),AΘ,A{F(x;θ):θ∈Θ}XF:3?(1)22HF3{N(μ,σ):θ∈Θ},9Θ={(μ,σ):−∞<μ<∞,σ2>0}(2)0}(4).F3{R(a,b):−∞0}?GammaF31.Z&EX;I!:λαα−1−λxf(x:α,λ)=xe,x>0,Γ(α)AXN$FGammaFAGa(α,λ),9α>0P/λ>0k!{Ga(α,λ):α>0,λ>0}AGammaF3?2.iXtit−α(1)g4:ϕ(t)=Ee=(1−)λαα(2)E(X)=D(X)=λλ2(3)hα=1fGa(1,λ)8λ1F?20\nn122Ga(,)An(!χFAχ(n)2221n−1−x(n)22FχFI!:f(x,n)=nnxe,x>22Γ()20(!1FE(4)=dX1∼Ga(α1,λ),X2∼Ga(α2,λ),\X1X2EGFX1+X2∼Ga(α1+α2,λ)21FExp(λ)χ(n)FGammaFguvxU;=d?1X∼Ga(α,λ),9α>0,λ>0Dn(X1,···,Xn)(;=k=i=1XiFI!:f-GammaF;=d\XiF)nFi=1Xi∼Ga(nα,λ)λ2X∼Ga(α,λ),Y=kX,0qY∼Ga(α,k),k>0O\kX¯Ff-X∼Ga(α1,λ)BI!:λαα−1−λxf(x:α,λ)=xe,x>0,Γ(α)21\n-Y=kX,X=Y/k,X=1/k,-bλαyyα−1−λ1f(y:α,λ)=()ekΓ(α)kk(λ)αkα−1−λ=yek,Γ(α)λ∼Ga(α,)kX¯=1nni=1xi∼Ga(nα,nλ)22113X∼N(0,σ),Y=X∼Ga(,).22σ2fhy>0fYF:2√√FY(y)=P{Y≤y}=P{X≤y}=P{−y≤X≤y}√√=FX(y)−FX(−y),9FI!:√√111−y−fY(y)=[fX(y)+fX(−y)]√=√y2e2σ2,2y2πσ1√11-Γ(2)=π,-lY∼Ga(2,2σ2).23.χFyZ/`(1)2222&Xi∼N(0,1)iid,Y=X1+X2+···+Xn∼χ(n)(2)2nxi−μ22&X∼N(μ,σ),Σi=1(σ)∼χ(n)0,b>0<?{Be(a,b):a>0,b>0}ABetaF3.2.aab(1)E(X)=,D(X)=a+b(a+b)2(a+b+1)(2)ha=b=1fBe(1,1)F8(0,1)O.FR(0,1)3.22&X1∼χ(n1),X2∼χ(n2),\X1,X2EGFX1/n1(1)F=∼F(n1,n2),9F(n1,n2)5?(!X2/n2n1,n2FF?X1n1n2(2)∼Be(,)X1+X2224.zCXn1n2(1)&X∼F(n1,n2),∼Be(,)9C=1+CX22n1/n2,twx?(2)22&X1∼χ(n1),X2∼χ(n2),\X1,X2EGFY1=X1+X2Y2=X1/X2EGF?YtF1.2ZEX∼N(0,1),Y∼χ(n),\X,Y23\nXEGFAET=Y/nN$Fn(!tFAT∼t(n).2.t(n)FI!:n+1Γ()22x−n+1t(x;n)=n√(1+)2,−∞2fE(T)=0,D(T)=n−2(2)2∼F(1,n).&X∼t(n),Xnn1(3)&X∼t(n),Y=∼Be(,).n+X2221x2−(4)limn→∞t(x;n)=√e22πtFirF(HFh1!F31.ZF={f(x;θ):θ∈Θ}F3&=(X1,···,Xn)I!:(<F^)f(x1,···,xn;θ)=5?.kf(x1,···,xn;θ)=a(θ)exp{Qj(θ)Tj(x1,···,xn)}h(x1,···,xn)j=1O\Mu{x:f(x1,···,xn;θ)>0}*`θ,AlF31!F3CA1324\n1:0q0}*1!F3-Mu{x:f(x;θ)>0}=(−θ,θ)`)θ.23XN$HFN(μ,σ),=@)n
n{y@)R.Ax=(x1,···,xn),%&=(X1,···,Xn)FI!1n12√n2f(x;μ,σ)=()exp{−2(xi−μ)}2πσi=12σ2n2√1nnμnμxi=()exp{−+x¯−}2σ2σ22σ22πσi=11nμ2nμ1n√n2=()exp{−2σ2}exp{σ2x¯−2xi},2πσ2σi=15zc1nμ2nμ2√n2a(μ,σ)=()exp{−2},Q1(μ,σ)=2,2πσ2σσ21n2Q2(μ,σ)=−2,T1(x)=¯x,T2(x)=xi,h(x)=12σi=125\n22-lHF3{N(μ,σ):−∞<μ<∞,σ>0}1!F3?hH;vH;1.Z&AX
4FI!:11−1f(x)=n1exp{−(x−a)B(x−a)}(2π)2|B|22−19B6|B|9^`?BB|R6AAXN$FvHF,CAX∼Nn(a,B).2.g4:AX∼Nn(a,B),9g4:1ϕ(t)=exp{iat−tBt}29t=(t1,···,tn)3.pn/){/)6X=(X1,···,Xn),Y=(Y1,···,Ym)AZ=(Zij)r×s R6AE(X)=(E(X1),···,E(Xn)),E(Z)=(E(Zij))r×s,26\nD(X)=E(X−E(X))(X−E(X))⎛⎞D(X1)Cov(X1,X2)···Cov(X1,Xn)⎜⎟⎜⎟⎜⎟⎜⎟⎜⎜Cov(X2,X1)D(X2)···Cov(X2,Xn)⎟⎟⎜⎟=⎜⎟,⎜⎜.........⎟⎟⎜...⎟⎜⎟⎜⎟⎝⎠Cov(Xn,X1)Cov(Xn,X2)···D(Xn)9D(Xi)Xi/)Cov(Xi,Xj)XiXj{/)?Cov(X,Y)=(Cov(Y,X))=E(X−E(X))(Y−E(Y))⎛⎞⎜Cov(X1,Y1)···Cov(X1,Ym)⎟⎜⎟⎜⎟=⎜⎜.........⎟⎟,⎜⎟⎜⎟⎜⎟⎝⎠Cov(Xn,Y1)···Cov(Xn,Ym)4.ρ=√Cov(Xi,Xj)ijD(XAXiXj)KE10i)D(Xj)CAE10?LFnnxin−xiP{X=x,···,X=x;p}=pi=1(1−p)i=11nnnxi=(1−p)n(p)i=1,1−pnT(x1,···,xn)=xi,h(x1,···,xn)=1,i=1npTg(T(x1,···,xn),p)=(1−p)()1−pP{X=x1,···,Xn=xn;p}=h(x1,···,xn)·g(T(x1,···,xn),p),n-+f=)T(X1,···,Xn)=Xipwi=1?4X∼N(μ,1),−∞<μ<+∞)0qXμw31\n0X1,···,Xn
4FI!nnf(x;θ)=(√1)nexp{−1(x−μ)2}iii=12π2i=1nn222b(xi−μ)=(xi−x)+n(x−μ)i=1i=1n1n1212`=(√)exp{−(xi−x)}exp{−n(x−μ)}2π2i=12n12h(x1,···,xn)=exp{−(xi−x)}2i=1g(T(x,···,x),μ)=(√1)nexp{−1n(x−μ)2}1n2π2-+f=)Xμw?3.&FnθAD2.FAT(X1,···,Xn)1θ
4w225.X∼N(μ,σ),θ=(μ,σ))A0n22(X,Xi)(μ,σ)
4w?i=10(X1,···,Xn)(;=9
4I!:nL(x1,···,xn;θ)=f(xi;θ)i=1n112=√nexp{−2(xi−μ)}(2πσ)2σi=1n2112nμnμ=√nexp{−2σ2xi+σ2x−2σ2}.(2πσ)i=132\nh(x1,···,xn)=1,n2√112nμnμg(T(x1,···,xn),θ)=nexp{−2σ2xi+σ2x−2σ2}(2πσ)i=1n2-+f=)T(X1,···,Xn)=(X,Xi)θ=i=12)(μ,σ
4w?T(X1,···,Xn)θ
4w*?n2Tiθiw?-l*?qX,Xici=12μ,σw?33