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 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), Y )= RR xy f(x,y)dxdy = 1 RR x2+y21,xy dxdy = 1 2, P(X+Y 2 3 4,X Y ) = RR x2+y23 4,xy f(x,y)dxdy = 1 RR 3 4x 2+y21,xy 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+t21 f(T1,T2)(t1,t2)dt1dt2 =2 ZZ t1,t20,t1I,t1+t21 e(t1+t2)dt1dt2.(2) ? u = t1,v = t1+ t2, 6?:? ?(2)? P(T1 I,T1+ T2 1) =2 ZZ uI,v1 evdudv =2 Z uI du Z + 1 evdv = | I | e.(3) ? P(S = 1) = e, ?(1)?(3)? P(T1 I | S = 1) = P(T1 I,S = 1) P(S = 1) =| I | . ? 12.?V?Vx,Vy,Vz,?Maxwell- Boltzman? fVx,Vy,Vz(vx,vy,vz) = 1 (2kT)3/2 exp ? v 2 x+ v 2 y+ v 2 z 2kT ? , vx,vy,vz +, ?K?Boltzman?,T?,?e.?x? ? ?. ?Vx,Vy,Vz? fVx,Vy,Vz(vx,vy,vz) = 1 (2kT)3/2 exp ? v 2 x+ v 2 y + v2 z 2kT ? = 1 (2kT)1/2 exp ? v2 x 2kT ? 1 (2kT)1/2 exp ? v2 y 2kT ? 1 (2kT)1/2 exp ? v2 z 2kT ? , vx,vy,vz +, ?,Vx,Vy,Vz?,?Vx,Vy,Vz?N(0,kT).? e = 1 2mE(V 2 x + V 2 y + V 2 z) = 3 2mkT, ? m = 2e 3kT ,.(1) ?x? ? mE(| Vx|) = m (2kT)1/2 Z + | vx| exp ? v2 x 2kT ? dvx = m (2kT)1/2 ? Z 0 vxexp ? v2 x 2kT ? dvx ?:?7 + Z + 0 vxexp ? v2 x 2kT ? dvx ? = 2m (2kT)1/2 Z + 0 vxexp ? v2 x 2kT ? dvx = m (2kT)1/2 Z + 0 exp ? v2 x 2kT ? d(v2 x) =m ?2kT ?1/2 . ?(1)? mE(| Vx|) = 2e 3 ? 2 kT ?1/2 . ? 13.?X1, ,Xn?,?.?T = n P i=1 Xi?(n,)?,? f(t) = n (n 1)!t n1et, t 0. ?T = n P i=1 Xi? gT(t) = gX1(t)n,(1) ?gX1(t)?X1?.?X1?,? gX1(t) = Z + 0 e(t)xdx = t, t , ?(1)? gT(t) = ? t ?n ,t .(2) ?(n,)? g(t) = n (n 1)! Z + 0 xn1e(t)xdx = n (n 1)! (n 1)! ( t)n = ? t ?n ,t . ?(2)?T = n P i=1 Xi?(n,)?.? 14.?X1?X2?1?2?Poisson?.? X1+ X2?,?X1+ X2= n?,X1?. 8?:? ?X P(),?X? gX(t) =E(etX) = + X i=0 eit i i! e =e + X i=0 (et)i i! = exp(et 1),t (,+), ?, X1+ X2? gX1+X2(t) =gX1(t)gX2(t) =exp1(et 1)exp2(et 1) =exp(1+ 2)(et 1),t (,+), ?X1+ X2 P(1+ 2).? P(X1+ X2= n) = (1+ 2)n n! e(1+2),n = 0,1,2, .(1) ? P(X1+ X2= n,X1= m) =P(X1= m,X2= n m) = P(X1= m)P(X2= n m) = m 1 m! e1 nm 2 (n m)!e 2, m,n = 0,1,2, ,m n, ?(1)? P(X1= m | X1+ X2= n) =P(X1 = m,X1+ X2= n) P(X1+ X2= n) = ? n m ? 1 1+ 2 ?m? 2 1+ 2 ?nm , m,n = 0,1,2, ,m n. ? 15.?X1,X2,?, NX1,X2,?, N? P(N = n) = (1 )n1,n = 1,2, ,0 1. ? N P i=1 Xi?. ?,?N = n?, X1,X2,? ?,?,?N = n?, n P i=1 Xi?(n,)? fY |N(y | n) = n (n 1)!t n1et, t 0. ?Y = N P i=1 Xi? fY(y) = X n=1 fY |N(y | n)P(N = n) ?:?9 = et X n=1 (1 )tn1 (n 1)! = ete(1)t = et,t 0. ?Y?.? 16.?X1,X2,?, P(X1= 1) = 1 2,N Xi,i = 1,2,? ?, 0 1.?Y = N P i=1 Xi?,?. ?1?,?n,? E(Y | N = n) =E( N X i=1 Xi| N = n) = E( n X i=1 Xi| N = n) =E( n X i=1 Xi) = nE(X1) = 0, E(Y 2 | N = n) =E( N X i=1 Xi)2| N = n = E( n X i=1 Xi)2| N = n =E( n X i=1 Xi)2 = n X j=1 n X i=1 E(XiXj) = n X i=1 E(X2 i) = n, E(Y 3 | N = n) =E( N X i=1 Xi)3| N = n = E( n X i=1 Xi)3| N = n =E( n X i=1 Xi)3 = n X j=1 n X i=1 n X k=1 E(XiXjXk) = 0, E(Y 4 | N = n) =E( N X i=1 Xi)4| N = n = E( n X i=1 Xi)4| N = n =E( n X i=1 Xi)4 = n X j=1 n X i=1 n X k=1 n X l=1 E(XiXjXkXl) = n X i=1 E(X4 i) + n X i,j=1,i6=j E(X2 iE(X 2 j) = n 2, ?Y? E(Y ) = EE(Y | N) = 0, ? E(Y 2) =EE(Y2 | N) = E(N) 10?:? = X n=1 n(1 )n1= 1 , ? E(Y 3) = EE(Y3 | N) = 0, ? E(Y 4) =EE(Y4 | N) = E(N2) = X n=1 n2(1 )n1= 1 . ?2?,?n,? E(etY| N = n) =E(expt N X i=1 Xi | N = n) = E(expt n X i=1 Xi | N = n) =E(expt n X i=1 Xi) = E(etX1)n = ? 1 2(e t + et) ?n , ? E(etY| N) = ?1 2(e t + et) ?N , ?Y? gY(t) =E(etY) = EE(etY| N) = X n=1 ? 1 2(e t + et) ?n (1 )n1 = (et+ et) 2 (1 )(et+ et). ?Y? E(Y ) = dgY(t) dt |t=0= 0, ? E(Y 2) = d2gY(t) dt2 |t=0= 1 , ? E(Y 3) = d3gY(t) dt3 |t=0= 0, ? E(Y 4) = d4gY(t) dt4 |t=0= 1 . ?:?11 ? 17.?N?Poisson?.?N = n,?M?n ?p?.?M?. ? P(M = m | N = n) = ? n m ? pm(1 p)nm,m = 0,1,2, ,n,n = 0,1,2, , P(N = n) = ne n! ,n = 0,1,2, , ?M? P(M = m) = X n=m P(M = m | N = n)P(N = n) = pme X n=m ? n m ?(1 p)nmn n! = (p)me m! X n=0 (1 p)n n! = (p)mep m! ,m = 0,1,2, . ?Y?p?Poisson?.? 12?:? ?2 1.?N(t) : t 0?Poisson?,?0 s 0. P(N(s) = k,N(t) = n) = P(N(s) = k,N(t) N(s) = n k) = P(N(s) = k)P(N(t) N(s) = n k) = (s)k k! es (t s)nk (n k)! e(ts) = nsk(t s)nk k!(n k)! et, k = 0,1, ,n,n = 0,1,2, ,0 s t. ? P(N(s) = k | N(t) = n) = P(N(s) = k,N(t) = n) P(N(t) = n) = n! k!(n k)! ?s t ?k? 1 s t ?nk , k = 0,1, ,n,n = 0,1,2, ,0 s t) = P(N(t) = N(4) = e3(t4),t 4. ?T1? FT1(t) = 1 P(T1 t) = 1 e3(t4),t 4, ? fT1(t) = dFT1(t) dt = 3e3(t4),t 4. ?T1 4?3?. ? 4.N(t) : t 0? = 2?Poisson?,? (i)P(N(1) 2); (ii)P(N(1) = 1?N(2) = 3); (iii)P(N(1) 2 | N(1) 1). ?(i) P(N(1) 2) = P(N(1) = 0) + P(N(1) = 1) + P(N(1) = 2) = e2+ 21 1! e2+ 22 2! e2 0.6767. (ii) P(N(1) = 1?N(2) = 3) = P(N(1) = 1,N(2) N(1) = 2) = P(N(1) = 1)P(N(2) N(1) = 2) = 21 1! e2 2(2 1)2 2! e2(21) 0.0733. (iii) P(N(1) 2 | N(1) 1) = P(N(1) 2,N(1) 1) P(N(1) 1) = P(N(1) 2) P(N(1) 1) = 1 P(N(1) = 0) P(N(1) = 1) 1 P(N(1) = 0) = 1 P(N(1) = 1) 1 P(N(1) = 0) = 1 21 1!e 2 1 e2 0.6870. 14?:? ? 6.?600?240?,?Poisson? 3?k?,k = 0,1,2, ,240. ? P(N(m + 3) N(m) = k | N(600) = 240), k = 0,1,2, ,240,m = 0,1,2, ,597.(1) ? P(N(m + 3) N(m) = k,N(600) = 240) = 240k X l=0 P(N(m) = l,N(m + 3) N(m) = k,N(600) N(m + 3) = 240 k l) = 240k X l=0 P(N(m) = l)P(N(m + 3) N(m) = k)P(N(600) N(m + 3) = 240 k l) = 240k X l=0 (m)l l! em (3)k k! e3 (597 m)240kl (240 k l)! e(597m) =240e600 3k k! 240k X l=0 ml l! (597 m)240kl (240 k l)! =240e600 3k597240k k!(240 k)!, k = 0,1,2, ,240,m = 0,1,2,597, P(N(600) = 240) = (600)240 240! e600, ?,?(1)? P(N(m + 3) N(m) = k | N(600) = 240) = P(N(m + 3) N(m) = k,N(600) = 240) P(N(600) = 240) = 240! k!(240 k)! ? 3 600 ?k? 1 3 600 ?240k , k = 0,1,2, ,240,m = 0,1,2, ,597.(2) ?: lim n,npn n! k!(n k)!p k n(1 pn) nk = k k! e,k = 0,1,2,(3) ?(2)? P(N(m + 3) N(m) = k | N(600) = 240) (240 3/600) k k! e2403/600= (6/5)k k! e6/5, ?:?15 k = 0,1,2, ,240,m = 0,1,2, ,597.(4) ?(2)?(4)? k?k? 00.30030.301260.00120.0012 10.36220.361470.00020.0002 20.21750.216980.00000.0000 30.08670.086790.00000.0000 40.02580.0260100.00000.0000 50.00610.0062110.00000.0000 ? 8.?Ni(t) : t 0,i = 1, ,n?n?i,i = 1, ,n? Poisson?,?T?n?,?T?. ? T t = Ni(t) = 0,i = 1, ,n,t 0, ? P(T t) = P(Ni(t) = 0,i = 1, ,n) = P(N1(t) = 0)P(Nn(t) = 0) = e t n P i=1 i, t 0, ?, T? FT(t) = 1 P(T t) = 1 e t n P i=1 i, t 0, ? fT(t) = dFT(t) dt = n X i=1 i e t n P i=1 i, t 0. ?T? n P i=1 i?.? 8.?N(t) : t 0?Poisson?.?N(t) = n,?r? ?Wr?fWr|N(t)(wr| n),r = 1, ,n. ? Wr wr = N(wr) r,r = 1, ,n, ? P(Wr wr| N(t) = n) = P(N(wr) r | N(t) = n) 16?:? = P(N(wr) r,N(t) = n) P(N(t) = n) ,wr,t 0.(1) ?0 wr t?,? P(N(wr) r,N(t) = n) = n X k=r P(N(wr) = k,N(t) = n) = n X k=r P(N(wr) = k,N(t) N(wr) = n k) = n X k=r P(N(wr) = k)P(N(t) N(wr) = n k) = n X k=r (wr)k k! ewr (t wr)nk (n k)! e(twr) =net n X k=r wk r(t wr) nk k!(n k)! ,r = 1, ,n.(2) ? P(N(t) = n) = (t)n n! et,n = 0,1,2, .(3) ?(2)?(3)?(1)? P(Wr wr| N(t) = n) = 1 tn n X k=r n! k!(n k)!w k r(t wr) nk, 0 wr t,r = 1, ,n.(4) ? Z wr 0 ur1(t u)nrdu =1 r Z wr 0 (t u)nrd(ur) =1 r wr r(t wr) nr + n r r Z wr 0 ur(t u)nr1du = =(r 1)!(n r)! n X k=r wr r(t wr) nr k!(n k)! , 0 wr t,r = 1, ,n, ?(4)? P(Wr wr| N(t) = n) = n! (r 1)!(n r)!tn Z wr 0 ur1(t u)nrdu, 0 wr t,r = 1, ,n. ?,?N(t) = n,Wr? fWr|N(t)(wr| n) = n! (r 1)!(n r)!tn wr1 r (t wr)nr,0 wr t,r = 1, ,n. ?:?17 ? 9.?Poisson?N(t) : t 0,?p? ?,?N1(t) : t 0.?N1(t) : t 0? N(t) N1(t) : t 0?N1(t) : t 0N(t) N1(t) : t 0? ? (i)N1(0) = 0; (ii)N1(t) : t 0?. ? (iii)?0 s t, N1(t) N1(s)?p(t s)?Poisso?.?,? (i)(iii)?, N1(t) : t 0?p?Poisso?.?,?0 s t,? P(N1(t) N1(s) = m | N(t) N(s) = n) = ? n m ? pm(1 p)nm, m = 0,1, ,n,n = 0,1,2, .(1) ?N(t) : t 0?Poisson?,?0 s t,? P(N(t) N(s) = n) = (t s)n n! e(ts),n = 0,1,2, .(2) ?(1)?(2)? P(N1(t) N1(s) = m) = X n=m P(N1(t) N1(s) = m | N(t) N(s) = n)P(N(t) N(s) = n) = X n=m ? n m ? pm(1 p)nm (t s)n n! e(ts) =p m m! e(ts) X n=m (1 p)nm(t s)n (n m)! =p(t s) m m! ep(ts),m = 0,1, , ?,?0 s t, N1(t) N1(s)?p(t s)?Poisso?. ?n?A1, ,An?,?n? A1, , An?,? Ai ?Ai?Ai?,i = 1, ,n,?,N1(t) : t 0N(t)N1(t) : t 0 ?.?N1(t)?N(t)N1(t)?, N(t)N1(t) : t 0?(1p) ?Poisso?.? 10.?1?Poisson?N1(t) : t 0,? ?2?Poisson?N2(t) : t 0,?N1(t) : t 0 N2(t) : t 0?.?N(t) : t 0 =N1(t)+N2(t) : t 0? ?N(t) : t 0?,?. ?,?,N(0) = 0. 18?:? ?,?N1(t) : t 0N2(t) : t 0?,?,?0 t1 tn,n ? ?(N1(t1), ,N1(tn)(N2(t1), ,N2(tn)?,?, n 1? ? (N1(t2) N1(t1), ,N1(tn) N1(tn1)(N2(t2) N2(t1), ,N2(tn) N2(tn1)? ?,?N1(t2)N1(t1), ,N1(tn)N1(tn1)?, N2(t2)N2(t1), ,N2(tn)N2(tn1) ?,?N1(t2) N1(t1), ,N1(tn) N1(tn1),N2(t2) N2(t1), ,N2(tn) N2(tn1) ?,?,N(t2) N(t1), ,N(tn) N(tn1)?.?,N(t) : t 0? ?. ?,?0 s 0, ? P(T t) = P(X(t) ) = P( N(t) X k=1 Yk ) = X n=1 P( N(t) X k=1 Yk ,N(t) = n) = X n=1 P( N(t) X k=1 Yk | N(t) = n)P(N(t) = n) = X n=1 P( n X k=1 Yk | N(t) = n)P(N(t) = n) ?:?19 = X n=1 P( n X k=1 Yk )P(N(t) = n),t 0.(1) ?Yk,k = 1,2,?, Y1?,?,Sn = n P k=1 Yk? ? fSn(s) = n (n 1)!s n1es, s 0,n = 1,2, , ? P( n X k=1 Yk ) = n (n 1)! Z 0 sn1esds,n = 1,2, .(2) ? P(N(t) = n) = (t)n n! et,n = 0,1,2, .(3) ?(2)?(3)?(1)? P(T t) = et X n=1 (t)n n!(n 1)! Z 0 sn1esds,t 0. ? E(T) = Z + 0 P(T t)dt = X n=1 ()n n!(n 1)! Z + 0 tnetdt Z 0 sn1esds = 1 X n=1 n (n 1)! Z 0 sn1esds = 1 Z 0 es X n=1 n (n 1)!s n1ds = Z 0 ds = . ?,?(?),?(? ?), ?(?),?.? ?.? 12.?N(t) : t 0?(t)?Poisson?,X1,X2,? ?. (i)X1,X2,? (ii)X1,X2,? (iii)?(X1,X2)?. ? Wk t = N(t) k,k = 1,2, , 20?:? ?,(W1,W2)? F(W1,W2)(t1,t2) = P(W1 t1,W2 t2) = P(N(t1) 1,N(t2) 2) = X k=2 k X l=1 P(N(t1) = l,N(t2) = k),0 t1 t2.(1) ?1 l k,0 t1 t2,? P(N(t1) = l,N(t2) = k) = P(N(t1) = l,N(t2) N(t1) = k l) = P(N(t1) = l)P(N(t2) N(t1) = k l) = m(t1)l l! em(t1) m(t2) m(t1)kl (k l)! em(t2)m(t1) = m(t1)lm(t2) m(t1)kl l!(k l)! em(t2), ?(1)? F(W1,W2)(t1,t2) = em(t2) X k=2 1 k! k X l=1 ?k l ? m(t1)lm(t2) m(t1)kl = em(t2) X k=2 1 k! ( k X l=0 ?k l ? m(t1)lm(t2) m(t1)kl m(t2) m(t1)k ) = em(t2) X k=2 1 k! ?m(t 2)k m(t2) m(t1)k ? = em(t2) ( X k=