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西电随机过程冯海林课后习题答案.pdf

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    5ŧ-Ž†A^6©‰Y Dean —————————————————————————- V¡‹< #¥ ··· —————————————————————————- 2012.12
    1 ŧģ ‡Eѧ,=§ŒǑ 1. Y = {Yt : t ≥ 0} ëêǑ lѧ…†Õ Ó©ÅCS x 1, . . . ,x n, . . . ¥ N = {Nt : t ≥ 0} ƒÕ .b µ = Ex 1,s 2 = D(x 1)Ñk,ŽEѧ Yƒê RY (s,t)Æ ê CY (s,t). ÄkŠê mY (t)ê DY (t).5¿ x k, Y0 = 0 Yt = k=1 Nt P{Nt = n} n ix i=1 n=0n(µ2 + s 2) + (n2 − n)µ2 (l t)n n! e−l t E(Y 2 = = mY (t) = n=0 k=1 n! = n=0 n! n n=1 k=1 k=1 n ix n=0 k=1 n=0 i=1 j=1 E(x ix j) E(x 2 i ) + x k!2 i, j=1 e−l t = E(x k) (l t)n n! e−l t = l µt, Nt = n  (l t)n e−l t n! Nt = n# (l t)n E" Nt x k e−l t = x k!2  t ) = E E Nt E  n  Nt    j# P{Nt = n} = E" n j)! (l t)n n=0 n n E(x i6= j = s 2l t + µ2(l t)2 + l t. Ïd 2ƒêÆê.Œ±y² YǑ²Õ §, „/ ? t − mY (t)2 = l (s 2 + µ2)t. RY (s,t) = l (s 2 + µ2)(s∧ t) + l 2µ2st. mY (t) = l µt, DY (t) = EY 2 RY (s,t) = E[Ys(Yt −Ys +Ys)] s ) + EYsE(Yt −Ys) s≤t = E(Y 2 = DY (s) + mY (s)2 + mY (s)mY (t − s) = l (s 2 + µ2)s + l 2s2µ2 + l sµl (t − s)µ = l (s 2 + µ2)s + l 2µ2st. CY (s,t) = RY (s,t)− mY (s)mY (t) = l (s 2 + µ2)(s∧ t). 1 ¥ ¥ ¥ ¥ x ¥ ¥ ¥
    · 2· t ≥ 0. . ... ... , Yt = µt + s Wt , ,Ï   (Ws1, . . .,Wsn) = (Ws1 −W0,Ws2 −Ws1, . . . ,Wsn −Wsn−1) 1 1 1··· 1 1 0 1 1··· 1 1 0 0 1··· 1 1 ... ... ... ... 0 0 0··· 1 1 0 0 0··· 0 1 ŧģ 2. W = {Wt : t ≥ 0} Ä.?~ê µ ∈ R,s > 0,  Ǒ (µ,s 2)-Ä.y²§d§,¿Ñ (Yt1, . . . ,Ytn)Š ¡ Y = {Yt : t ≥ 0} Æ,¥ 0 < t1 < ··· < tn < ¥ Äk`²Ä WǑd§. ∀ 0 ≤ s0 < s1 < ··· < sn < ¥ dÄ Â9C†5Ÿ (P.14) WǑd§.gy² YǑd  §.Ï 2g^C†5Ÿ YǑd§.w Æ CCC = (ci j)n×n,¥ ci j = s 2(ti ∧ t j), ¤  5. x h‡ƒÕ ÅC,¥ xVÇê h ∼ U [0, 2p ].y²Å§ E(Yt1, . . . ,Ytn) = (EYt1, . . . , EYtn) = (t1µ, . . .,tnµ), Cov(Yti,Yt j) = s 2E(WtiWt j) = s 2(ti ∧ t j), t1 t2 t3 ... t1 t1 t1 ··· t1 t2 t2 ··· t1 t2 t3 ··· ... ... ... t1 t2 t1 t2 t1 t2 t3 ... ... t3 ··· tn−1 tn−1 t3 ··· tn−1 tn (Ys1, . . . ,Ysn) = s (Ws1, . . .,Wsn) + (µs1, . . . , µsn), fx (x) = 2x3e−x4/21(0,¥ )(x), CCC = s 2     . Xt := x 2 cos(2p t + h ), t ≥ 0
    · 3· d dx √x d dx Z 0 P{x 2 ≤ x} = 2t3e−t4/2dt = xe−x2/2, x > 0, , X2 = x 2 sinh Xt = (X1, X2)(cos(2p t),−sin(2p t))T . (Xt1, . . ., Xtn) = (X1, X2) cos(2p t1) cos(2p t2) ··· cos(2p tn) −sin(2p t1) −sin(2p t2) ··· −sin(2p tn)! , ŧģ ‡§,¿ŽŠêÆê. Ï x 2ÑëêǑ s 2 = 1 Rayleigh© (P.31). X1 = x 2 cosh , d~ 2.2.5 (X1, X2)Ñ©, XtÑ©.?é?Óž  t1, . . . ,tn ∈ [0,¥ ) (Xt1, . . ., Xtn)Ñ n©,ùÒy² XǑ§.   Ïd ∀ s,t ∈ [0,¥ ),dÕ 5 6.ŽÄ W©ê9VÇê.  ∀ 0 ≤ s < t,w (Xs, Xt)Ñ©,ŠǑ µµµ = (0, 0),ÆǑ VÇê (P.14)Ǒ mX (t) = E[x 2 cos(2p t + h )] = E(x 2)Ecos(2p t + h ) = 0, RX (s,t) = E(x 4)E[cos(2p s + h ) cos(2p t + h )] = 2· CX (s,t) = RX (s,t)− mX(s)mX(t) = cos[2p (t − s)]. Ecos(2p t + h ) =Z 2p E[cos(2p s + h ) cos(2p t + h )] =Z 2p E(x 4) =Z −¥ cos[2p (t − s)] = cos[2p (t − s)], t! , CCC−1 = cos(2p s + u) cos(2p t + u) cos[2p (t − s)], 1 2p du = 0, CCC = s s x4 fx (x)dx = 2. 1 2p du = 1 2 cos(2p t + u) 1 2 0 0 s 1 1 2 s ! . −s st − s2 t −s (xxx− µµµ)CCC−1(xxx− µµµ)T s ! (x1, x2)T) (x1, x2) t −s 2(st − s2) −s x2 1t + x2 2s− 2x1x2s , 2(st − s2) 1 f (x1, x2)) = = = 1 1 (2p )n/2|CCC|1/2 exp− exp(− 2p √st − s2 exp− 2p √st − s2 1 ¥
    · 4· F(x, y) = f (x1, x2)dx1dx2. ŧģ ©êǑ 9. N = {Nt : t ≥ 0} ‡ëêǑ l > 0ѧ.é ~ê l > 0 Â#§ Šê mZ(l)ƒê RZ(l)(s,t)9ƒê RN,Z(l)(s,t). Ž Z(l) = {Z(l) Äkѧƒê. s < t,dÕ 59Ñ©k „/ 2 Z(l)Šꆃê.w RN(s,t) = E[(Nt − Ns + Ns)Ns] = ENsE(Nt − Ns) + E(N2 RN(s,t) = l (s∧ t) + l 2st. Z(l) t = Nt+l − Nt, s ) = l sl (t − s) + l s + (l s)2 = l s + l 2st. x y Z −¥ Z −¥ t : t ≥ 0} t ≥ 0. mZ(t) = E(Nt+l − Nt ) = l (t + l)− l t = l l; RN,Z(l)(s,t) = E[Ns(Nt+l − Nt)] = RN(s,t + l)− RN(s,t) = l 2s(t + l) + l [s∧ (t + l)]− [l 2st + l (s∧ t)] = l 2sl + l [s∧ (t + l)− s∧ t]; RZ(l)(s,t) = E[Ns+lNt+l − NtNs+l − NsNt+l + NsNt ] = RN(s + l,t + l)− RN(t, s + l)− RN(s,t + l) + RN(s,t) = l 2l2 + l [(s + l)∧ (t + l)− t ∧ (s + l)− s∧ (t + l) + s∧ t].
    i) ii) iii) −W0 = −0 = 0. −(Wt1 −Wt0),−(Wt2 −Wt1), . . . ,−(Wtn −Wtn−1). 1 Ä 1.yÄ Wé¡5,gƒ5ž_=5. y? 0 ≤ t0 < t1 < ··· < tn−1 < tn < ¥9 t ≥ 0. −WǑÄ. (1)é¡5=y ("Š)w (²Õ ) −WSǑ Ï Wtk − Wtk−1 ∼ N(0,tk − tk−1), −(Wtk − Wtk−1) ∼ N(0,tk − tk−1),= −W÷v ²5. Wt1 − Wt0,Wt2 − Wt1, . . . ,Wtn − Wtn−1 ƒÕ , −WǑ÷vÕ 5. Wt1), . . .,−(Wtn −Wtn−1)ǑwƒÕ ,`² ()w Œ, −WǑÄ. (2)gƒ5=y ∀ t, a > 0,k Wat Ï Wat ∼ N(0, at),d©5Ÿ √aWt ∼ N(0, at), Wat (3)ž_=5=yé T > 0,§ B = {Bt = WT −WT−t : 0 ≤ t ≤ T}  Ä. ("Š)w B0 = WT −WT = 0. (²Õ ) Btk − Btk−1 = WT−tk−1 −WT−tk, k = 1, . . ., n,tn ≤ T , BSǑ w 0 ≤ T − tn < T − tn−1 < ··· < T − t1 < T − t0 ≤ T ,a (1)¥ ii)?é BÕ 5. ()d W  WT −WT−t ∼ N(0,t),= Bt ∼ N(0,t). Œ, BǑÄ. t ),¥ t > 0, mǑê. 2.ŽÄ W m E(W m WT−t0 −WT−t1,WT−t1 −WT−t2, . . .,WT−tn−1 −WT−tn. −(Wt1 − Wt0),−(Wt2 − i) ii) iii) −Wt ∼ N(0,t). = √aWt. d = √aWt. d 5
    0, k=0 t ≥ 0. · 6· E(W m Ck k=0 E[W 4 t −W 4 E[W 3 t −W 3 Cm−k m xm−kµk. s − 6(t − s)W 2 s − 3(tWt − sWs)] kǑóê; kǑê. µk := E[(Wt − x)k] =((k− 1)!!tk/2, mxm−k(Wt − x)k# = t ) = E" m m Ä  WŠ W0 = x,Ï Wt ∼ N(x,t),d P.13 3. W‡Ä, ∀ s < t,  dÏ"‚55Ÿ ªÑǑ 0. 6. W‡Ä,é t0 > 0,  y² W 0 = {W 0 ǑǑÄ. © ("Š)Ï W 0 0 = Wt0 −Wt0 = 0, W 0ŠǑ 0. (²Õ ) ∀ 0 ≤ s0 < s1 < ··· < sn < ¥ ,d W²5k W 0²§.d WÕ §S ƒÕ , ƒÕ ,= W 0÷vÕ 5. ()Ï W 0 si−1 = Wsi+t0 −Wsi−1+t0 ∼ N(0, si − si−1), W 0 t = Wt+t0 −Wt0, s − 3(t − s)2]. W 0 s1 −W 0 t = Wt+t0 −Wt0 ∼ N(0,t),=§ W 0©. sn −W 0 sn−1 sn−1 −W 0 sn−2,W 0 s1, . . .,W 0 s2 −W 0 s0,W 0 W 0 si −W 0 t : t ≥ 0} i = 1, . . . , n, (1) (2) (3) Ws1+t0 −Ws0+t0,Ws2+t0 −Ws1+t0, . . .,Wsn−1+t0 −Wsn−2+t0,Wsn+t0 −Wsn−1+t0
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