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n:^! G9 0 |- W9’1 XVO jf\Qie11.R{X(t),t∈T}IY%3; F1 }., (6 pQFC{*CE[X(s)]ÆE[X(s)X(s+t)]P }A s.o pQ}.FM *,{X(t),t∈T}pQC{*CZOK!A ,G:(1)μX(s)=E[X(s)] s∈T ’&;(2)RX(t,s)=Cov[X(t),X(s)]0Æt−s x.; DCov[X(t),X(s)]=E[X(t)X(s)]−E[X(t)]E[X(s)], 8,!(1) (2)ÆZOK!H :(3)μX(s)=E[X(s)] s∈T ’&;(4)E[X(t)X(s)]0Æt−s x.#1(dN{X(t),t∈T}pQC{*CE[X(s)]ÆE[X(s)X(s+t)]P }A s.2. U1,···,UnI (0,1)57 b FnnQG1 M.V0t,x1,M I(t,x)=(1,x≤t,0,xt, X(t)=1nnPk=1I(t,Uk),0≤t≤1,# U1,···,UnF.wb &. }1 }.{X(t),0≤t≤1}F7- Bf_% &.[ : *,{X(t),0≤t≤1}F7- &IμX(t)=E[X(t)]=1nnXk=1E[I(t,Uk)]=E[I(t,U1)],0≤t≤1,(1)Bf_% &IRX(t,s)=Cov[X(t),X(s)]]=1n2nXk=1nXl=1Cov[I(t,Uk),I(s,Ul)]=1n2nXk=1Cov[I(t,Uk),I(s,Uk)]=1nCov[I(t,U1),I(s,U1)],0≤t≤1,(2)XE[I(t,U1)]=P(U1≤t)=t,0≤t≤1,(3)2 n:^! G9 0 |- W9’E[I(t,U1)I(s,U1)]=P(U1≤t,U1≤s)=P(U1≤min{t,s})=min{t,s},0≤t,s≤1,Cov[I(t,U1),I(s,U1)]=E[I(t,U1)I(s,U1)]−E[I(t,U1)]E[I(s,U1)]=min{t,s}−ts,0≤t,s≤1,(4)(3) (4) (1) (2)5EμX(t)=t,0≤t≤1,RX(t,s)=1n[min{t,s}−ts],0≤t,s≤1.3.RZ1,Z2IQGAb F’81 M,7-I0,_%Iσ2,λI &.M X(t)=Z1cos(λt)+Z2sin(λt). }{X(t),t∈(−∞,+∞)}F7- & Bf_% &.6 pQFZ?[ : *,{X(t),t∈(−∞,+∞)}F7- &IμX(t)=E(Z1)cos(λt)+E(Z2)sin(λt)=0,t∈(−∞,+∞),Bf_% &IRX(t,s)=Cov[Z1cos(λt)+Z2sin(λt),Z1cos(λs)+Z2sin(λs)]=Cov(Z1,Z1)cos(λt)cos(λs)+Cov(Z1,Z2)cos(λt)sin(λs)+Cov(Z2,Z1)sin(λt)cos(λs)+Cov(Z2,Z2)sin(λt)sin(λs)=σ2cos(λt)cos(λs)+σ2sin(λt)sin(λs)=σ2cos[λ(t−s)],t,s∈(−∞,+∞).w{X(t),t∈(−∞,+∞)} pQF.4.Poisson}.{X(t),t≥0}[F(i)X(0)=0;(ii)Vts,X(t)−X(s)f:7-Iλ(t−s)FPoissonb ;(iii)}. QG MF. }r7- & Bf_% &.6 pQFZ?[{X(t),t≥0}F7- &IμX(t)=E[X(t)]=E[X(t)−X(0)]=λt,t0,{X(t),t≥0}F_% &IVar[X(t)]=Var[X(t)−X(0)]=λt,t0,Vst0, E[X(t)X(s)]=E{[X(t)−X(0)][(X(s)−X(t))+(X(t)−X(0))]} n:^! G9 0 |- W9’3=E{[X(t)−X(0)]2}+E{[X(t)−X(0)][X(s)−X(t)]}=Var[X(t)−X(0)]+{E[X(t)−X(0)]}2+E[X(t)−X(0)]E[X(s)−X(t)]=λt+(λt)2+λt·λ(s−t)=λt(λs+1), 8,{X(t),t≥0}FBf_% &IRX(t,s)=E[X(t)X(s)]−E[X(t)]E[X(s)]=λt,s≥t0,B‘x &IrX(t,s)=RX(t,s)[RX(t,t)RX(s,s)]1/2=rts,s≥t0.5.{X(t),t≥0}IJ*:5FPoisson}.. Y(t)=X(t+1)−X(t), }}.{Y(t),t≥0}F7- & Bf_% &, u0rpQm.[{Y(t),t≥0}F7- &IμY(t)=E[X(t+1)]−E[X(t)]=μX(t+1)−μX(t)=λ,t≥0,Bf_% &IRY(t,s)=Cov[X(t+1)−X(t),X(s+1)−X(s)]=Cov[X(t+1),X(s+1)]−Cov[X(t+1),X(s)]−Cov[X(t),X(s+1)]+Cov[X(t),X(s)]=λ(min{t,s}+1)−λmin{t+1,s}−λmin{t,s+1}+λmin{t,s},=0,C0≤ts−1,λ(t−s+1),Cs−1≤ts,λ(s−t+1),Cs≤ts+1,0,Ct≥s+1,t,s≥0.#)dN{Y(t),t≥0} pQF.6.RZ1 Z2 QGAb F1 M,P(Z1=−1)=P(Z1=1)=12. X(t)=Z1cos(λt)+Z2sin(λt),t∈R, ({X(t),t∈R} pQF,6 tpQFZ?o‘ : *,{X(t),t∈R}F7- &IμX(t)=E(Z1)cos(λt)+E(Z2)sin(λt)=0,t∈R,(1)4 n:^! G9 0 |- W9’Bf_% &IRX(t,s)=Cov(Z1cos(λt)+Z2sin(λt),Z1cos(λs)+Z2sin(λs))=Var(Z1)cos(λt)cos(λs)+Var(Z2)sin(λt)sin(λs)=cos(λt)cos(λs)+sin(λt)sin(λs)=cos(λ(t−s)),t,s∈R.(2) (1) (2) *,{X(t),t∈R} pQF.+X, : *,1 MX(t)F3h &IgX(t)(u)=E(euX(t))=E{exp[u(Z1cos(λt)+Z2sin(λt))]}=E{exp[uZ1cos(λt)]}·E{exp[uZ2sin(λt)]}=14{exp[−ucos(λt)]+exp[ucos(λt)]}·{exp[−usin(λt)]+exp[usin(λt)]},u∈R,#)dNX(t)Fb Æt(∈R) x, 8{X(t),t∈R} tpQF.7. (: Z0,Z1,Z2,···IQGAb 1 MpO,M X(n)=Z0+Z1+···+Zn,n=0,1,2,···, {X(n),n=0,1,2,···} QG M}..o‘; DV n t1,···,tn∈{0,1,2,···},t1t2···tn, X(t2)−X(t1)=Zt1+1+···+Zt2,X(t3)−X(t2)=Zt2+1+···+Zt3,············X(tn)−X(tn−1)=Ztn−1+1+···+Ztn.(1)X : *,Zt1+1,···,Ztn ‘QG, 8(Zt1+1,···,Zt2),(Zt2+1,···,Zt3),···,(Ztn−1+1,···,Ztn) ‘QG,w (1)*,X(t2)−X(t1),X(t3)−X(t2),···,X(tn)−X(tn−1) ‘QG.#1(dN{X(n),n=0,1,2,···} QG M}..8. X1,X2,··· |OQG1 M, y \!!k Æ{X1,X2,···} tpQF1 }.?[ {X1,X2,···}tpQ, V ’&&m n,Xm XnFb P‘A,:XX1,X2,··· |OAb F1 M.XCX1,X2,··· |OQGAb F1 M ,V ’&&k n1,···,nk,kJ1 M(Xn1,···,Xnk)Fb &I( X1,X2,···vAFb &IF(x))F(Xn1,···,Xnk)(x1,···,xk)=FXn1(x1)···FXnk(xn)=F(x1)···F(xk),−∞x1,···,xk+∞,#)dN(Xn1,···,Xnk)Fb &Æn1,···,nkSx,w{X1,X2,···}tpQ.9.RX Y :BN jF7 b 51 r |K3EF J EJ . /!kXPX2+Y2≥34|XY. n:^! G9 0 |- W9’5[; D(X,Y)FH bR &If(x,y)=(1π,x2+y2≤1.0,r4. 8P(XY)=RRxyf(x,y)dxdy=1πRRx2+y2≤1,xydxdy=12,P(X+Y2≥34,XY)=RRx2+y2≥34,xyf(x,y)dxdy=1πRR34≤x2+y2≤1,xydxdy=12h1− 342i=732,w3}!kXIPX2+Y2≥34|XY=P(X2+Y2≥34,XY)P(XY)=716.10.CA}&IλFPoissonb + &v,KCAD=F mT1,T2,··· QGF&IλF.&b 1 M. S [0,1] T5FCAD&, ~ I⊂[0,1],r(R I|I|. (dP(T1∈I,S=1)=P(T1∈I,T1+T21), 8 /P(T1∈I|S=1)=|I|.[o : *,JinCA T1+···+Ti =+ &v,i=1,2,···,XS [0,1] Tj+ &vFCAD&, 8{T1∈I,S=1}={T1∈I,T1+T21},wP(T1∈I,S=1)=P(T1∈I,T1+T21).(1)X(T1,T2)FH bR &If(T1,T2)(t1,t2)=fT1(t1)fT2(t2)=λ2e−λ(t1+t2),t1,t20, XP(T1∈I,T1+T21)=ZZt1∈I,t1+t21f(T1,T2)(t1,t2)dt1dt2=λ2ZZt1,t20,t1∈I,t1+t21e−λ(t1+t2)dt1dt2.(2)I u=t1,v=t1+t2,6 n:^! G9 0 |- W9’ (2)5 b IP(T1∈I,T1+T21)=λ2ZZu∈I,v1e−λvdudv=λ2Zu∈IduZ+∞1e−λvdv=λ|I|e−λ.(3)XP(S=1)=λe−λ, 8 (1) (3)EP(T1∈I|S=1)=P(T1∈I,S=1)P(S=1)=|I|.12.w;bAF.RV n7,bMVx,Vy,Vz,6‘FH bR &}Maxwell-BoltzmanMWIfVx,Vy,Vz(vx,vy,vz)=1(2πkT)3/2exp−v2x+v2y+v2z2kT,−∞vx,vy,vz+∞,r5K Boltzman’&,T 6VOR,pMbAFDNkIe. }x_FNMF6V-FqH-.[ Vx,Vy,VzFH bR &IfVx,Vy,Vz(vx,vy,vz)=1(2πkT)3/2exp−v2x+v2y+v2z2kT=1(2πkT)1/2exp−v2x2kT·1(2πkT)1/2exp−v2y2kT·1(2πkT)1/2exp−v2z2kT,−∞vx,vy,vz+∞, 8,Vx,Vy,Vz ‘QG,{Vx,Vy,VzPf:’8b N(0,kT).ww;bAFDNkIe=12mE(V2x+V2y+V2z)=32mkT, 8Em=2e3kT,.(1)Xw;x_FNMF6V-FqH-ImE(|Vx|)=m(2πkT