方兆本 随机过程 答案

?:?1 ? ? ?1 1.?X(t),t T?,?EX(s) EX(s)X(s + t)?s. ?,X(t),t T?: (1)X(s) = EX(s)?s T?; (2)RX(t,s) = CovX(t),X(s)?t s?. ? CovX(t),X(s) = EX(t)X(s) EX(t)EX(s), ?,?(1)?(2)?: (3)X(s) = EX(s)?s T?; (4)EX(t)X(s)?t s?. ?X(t),t T?EX(s)EX(s)X(s + t)?s.? 2.?U1, ,Un?(0,1)?n?.?0 < t,x t, ?X(t) = 1 n n P k=1 I(t,Uk),0 t 1,?U1, ,Un?.? X(t),0 t 1?. ?,X(t),0 t 1? X(t) = EX(t) = 1 n n X k=1 EI(t,Uk) = EI(t,U1),0 t 1,(1) ? RX(t,s) = CovX(t),X(s) = 1 n2 n X k=1 n X l=1 CovI(t,Uk),I(s,Ul) = 1 n2 n X k=1 CovI(t,Uk),I(s,Uk) = 1 nCovI(t,U1),I(s,U1), 0 t 1,(2) ? EI(t,U1) = P(U1 t) = t,0 t 1,(3) 2?:? EI(t,U1)I(s,U1) = P(U1 t,U1 s) = P(U1 mint,s) = mint,s,0 t,s 1, CovI(t,U1),I(s,U1) = EI(t,U1)I(s,U1) EI(t,U1)EI(s,U1) = mint,s ts,0 t,s 1,(4) ?(3)?(4)?(1)?(2)? X(t) = t,0 t 1, RX(t,s) = 1 nmint,s ts, 0 t,s 1. ? 3.?Z1,Z2?,?0,?2,?.? X(t) = Z1cos(t)+Z2sin(t).?X(t),t (,+)?. ? ?, X(t),t (,+)? X(t) = E(Z1)cos(t) + E(Z2)sin(t) = 0,t (,+), ? RX(t,s) =CovZ1cos(t) + Z2sin(t),Z1cos(s) + Z2sin(s) =Cov(Z1,Z1)cos(t)cos(s) + Cov(Z1,Z2)cos(t)sin(s) + Cov(Z2,Z1)sin(t)cos(s) + Cov(Z2,Z2)sin(t)sin(s) =2cos(t)cos(s) + 2sin(t)sin(s) =2cos(t s),t,s (,+). ?X(t),t (,+)?.? 4.Poisson?X(t),t 0?(i)X(0) = 0;(ii)?t > s,X(t) X(s)? (t s)?Poisson?;(iii)?.?.? ? ?X(t),t 0? X(t) = EX(t) = EX(t) X(0) = t,t > 0, X(t),t 0? V arX(t) = V arX(t) X(0) = t,t > 0, ?s > t > 0,? EX(t)X(s) =EX(t) X(0)(X(s) X(t) + (X(t) X(0) ?:?3 =EX(t) X(0)2 + EX(t) X(0)X(s) X(t) =V arX(t) X(0) + EX(t) X(0)2 + EX(t) X(0)EX(s) X(t) =t + (t)2+ t (s t) =t(s + 1), ?, X(t),t 0? RX(t,s) = EX(t)X(s) EX(t)EX(s) = t,s t > 0, ? rX(t,s) = RX(t,s) RX(t,t)RX(s,s)1/2 = r t s, s t > 0. ? 5.X(t),t 0?Poisson?.?Y (t) = X(t + 1) X(t),? Y (t),t 0?,?. ?Y (t),t 0? Y(t) = EX(t + 1) EX(t) = X(t + 1) X(t) = ,t 0, ? RY(t,s) =CovX(t + 1) X(t),X(s + 1) X(s) =CovX(t + 1),X(s + 1) CovX(t + 1),X(s) CovX(t),X(s + 1) + CovX(t),X(s) =(mint,s + 1) mint + 1,s mint,s + 1 + mint,s, = 0,?0 t < s 1, (t s + 1),?s 1 t < s, (s t + 1),?s t < s + 1, 0,?t s + 1, t,s 0. ?Y (t),t 0?.? 6.?Z1?Z2?, P(Z1= 1) = P(Z1= 1) = 1 2. ? X(t) = Z1cos(t) + Z2sin(t),t R,?X(t),t R?,? ?,X(t),t R? X(t) = E(Z1)cos(t) + E(Z2)sin(t) = 0,t R,(1) 4?:? ? RX(t,s) =Cov(Z1cos(t) + Z2sin(t),Z1cos(s) + Z2sin(s) =V ar(Z1)cos(t)cos(s) + V ar(Z2)sin(t)sin(s) =cos(t)cos(s) + sin(t)sin(s) =cos(t s),t,s R.(2) ?(1)?(2)?,X(t),t R?.?,?,?X(t)? ? gX(t)(u) =E(euX(t) = Eexpu(Z1cos(t) + Z2sin(t) =EexpuZ1cos(t) EexpuZ2sin(t) =1 4expucos(t) + expucos(t) expusin(t) + expusin(t),u R, ?X(t)?t( R)?,?X(t),t R?. ? 7.?:?Z0,Z1,Z2,?,?X(n) = Z0+Z1+ Zn,n = 0,1,2,?X(n),n = 0,1,2,?. ?n?t1, ,tn 0,1,2, t1< t2< < tn,? X(t2) X(t1) = Zt1+1+ + Zt2, X(t3) X(t2) = Zt2+1+ + Zt3, X(tn) X(tn1) = Ztn1+1+ + Ztn. (1) ?, Zt1+1, ,Ztn?,?(Zt1+1, ,Zt2),(Zt2+1, ,Zt3), ,(Ztn1+1, ,Ztn)?,?(1)?,X(t2) X(t1),X(t3) X(t2), ,X(tn) X(tn1)? ?.?X(n),n = 0,1,2,?.? 8.?X1,X2,?,?X1,X2,? ? ?X1,X2,?,?m?n, Xm?Xn?,? X1,X2,?.?X1,X2,?, ?k?n1, ,nk,k? ?(Xn1, ,Xnk)?(?X1,X2, ?F(x) F(Xn1,Xnk)(x1, ,xk) =FXn1(x1)FXnk(xn) =F(x1)F(xk), < x1, ,xk Y ? . ?:?5 ?(X,Y )? f(x,y) = ( 1 , x2+ y2 1. 0,?. ? P(X > Y )= RR x>y f(x,y)dxdy = 1 RR x2+y21,x>y dxdy = 1 2, P(X+Y 2 3 4,X > Y ) = RR x2+y23 4,x>y f(x,y)dxdy = 1 RR 3 4x 2+y21,x>y dxdy = 1 2 h 1 ?3 4 ?2i = 7 32, ? P ? X2+ Y 2 3 4 | X > Y ? = P(X2+ Y 2 3 4,X > Y ) P(X > Y ) = 7 16. ? 10.?Poisson?,?T1,T2, ?.?S?0,1?,? I 0,1,?| I |.?P(T1 I,S = 1) = P(T1 I,T1+ T2> 1),? ?P(T1 I | S = 1) =| I |. ?,?i?T1+Ti?,i = 1,2,?S? 0,1?,? T1 I,S = 1 = T1 I,T1+ T2> 1, ? P(T1 I,S = 1) = P(T1 I,T1+ T2> 1).(1) ?(T1,T2)? f(T1,T2)(t1,t2) = fT1(t1)fT2(t2) = 2e(t1+t2),t1,t2> 0, ? P(T1 I,T1+ T2> 1) = ZZ t1I,t1+t2>1 f(T1,T2)(t1,t2