隨機(jī)過(guò)程知識(shí)點(diǎn)匯總

X，F(xiàn)(x)P(Xx)P(Xx)F(x)pXpkkkx(x)()Fx()ftXf(X,X,,X)nX12n(x)F(x,x,,x)P(Xx,Xx,,Xx,)F12n1122nnxpXX(x)kkE(XEX)EX(EX)DX222,YB[(XYE()XYXB,Y0X,YX若0(t)E(e)gt)epgt)ef(x)gitxkkg(0)1，gt)1g(t)g(t)g(0)iEX，，kkkP(Xp,P(Xqp(Xk)CpqPkknknkP(Xk)eDXEXk!1(xa)22(a,)f(x)eEXa2N2,011exx(x)EXDXfx02(X,X,,X)~(,)XNaBX12n11f(x,x,,x)(xaB1(xa)T212nn1(2)|B|22a(a,a,,a)，x(x,x,,x)，B(b)12n12nnn(,P)tTX設(shè)TXt,e),tT是(,P)Xt),tT。t,e)eXt,e)當(dāng)tXt)TIXt),tTXnt)t)mD表示隨機(jī)過(guò)程Xt),tTtX(t)E[X(t)m(t)]2表示隨機(jī)過(guò)程在時(shí)刻tXXB(s,t)[(X(s)m(sXt)mtt,t)Dt)BXXX[X(s)Xt)]m(s)mt)XXXX(s,t)E[X(s)XtRs，tXt),tT，Yt),tTXB(s,t)[(X(s)m(sYt)mtXYXY(,)[()(RstEXsYtXY[X(sYt)]m(s)mt)XY若E[X(sYtm(s)mt)XYXjYZttt(t)EXjEYmZttDt)Zmt)|]2[(Zmt))(ZmtZtZtZtZB(s,t)[(Zm(s))(Zmt))](s,t)[ZZ]ZsZtZRZst[ZZ]m(s)mt)stZZXt),tTtTEXt)2t),tTttttTX1234[(Xt)XtXt)Xt02143(s,t)R(s,t)s,t))B2XXXt),tTXn2tttT，12nXt),tTXt)X(s)(t)X(t),X(t)X(t),,X(t)X(t)Xt2143nn1Xt),tTstXt),tTst),tT具有馬爾可夫性，即對(duì)任意正整數(shù)及（4）馬爾可夫過(guò)程：如果隨機(jī)過(guò)程XntttT，P(X(t)x,,X(t)x)012n11n1n1PXt)xXt)x,,Xt)xPXt)xXt)xXt),tTnn11n1n1nnn1n1t),tT（5）正態(tài)過(guò)程：隨機(jī)過(guò)程X，若對(duì)任意正整數(shù)nt,t,,tT及，12n(t),X(t)X(t)（Xnn12nXt),tT設(shè)Wt),tW(0)0s,t增t)W(s)()()~服從正態(tài)分布，即WtWsNts。則稱)0量W22Wt),t嚴(yán)（狹義）平穩(wěn)過(guò)程：Xt),tT，如果對(duì)任意常數(shù)和正整數(shù)及nt,t,,tT，12n,tt,,(),()()(),()()tTXtXtXtXtXtXt12n12n12nXt),tTXt),tTXt),tTtT，mt)t)s，tT，R(s,t)[X(s)XtRts)XXXs差t(t),tXXt),tT(0)0是具有的泊松①X；數(shù)n，tttTX(t)X(t),X(t)X(t),,X(t)X(t)12n2132nn1tt0(t)PXtsXsnent,s0()()ntn!E[X(t)]E[X(tt，tX(t),tX(t),t00是具有參數(shù)的泊松過(guò)程。①X；②獨(dú)立、平穩(wěn)增量過(guò)程；③PX(t)X(t)1ho()。(t)X(t)2o()PX(1)ststst(t)E[X(t)]t[X(t)](,)Rstm(1)tsXXB(s,t)R(s,t)m(s)m(t)min(s,t)XXXX,1Tn1n２，TnTnnn,01,0ettett(t)，分布函數(shù)F(t)f0,t00,t0n1為ETn略tPX(t)X(t)1(t)ho()①X0。(t)X(t)2o()PX(t)E[X(t(s)tmX0(t),tm(t)(s)XtX0[m(ts)m(t[m(ts)m(t)]nPX(ts)X(t)nXXn!XX設(shè)N(t),tY,kkN(t)N(t),t0X(t)YX(t),t0kk1X(t),t0EY)E[X(t)]tEY)重要結(jié)論：是獨(dú)立增量過(guò)程；若2，則，11[X(t)]tEY)21t0ttt00在有關(guān)，而與過(guò)去無(wú)關(guān)。表示為PXt)xXt)x,,Xt)xPXt)xXt)xnn11n1n1nnn1n11X,nTnTi,i,,iI01n1n,nTPXiXi,Xi,,XiPXiXiXn1n10011nnn1n1nnniXiPXn1n1nnjX2PXnin1nPX(n)p(n)ijjXinpjn1nij(n)pp與。nijijP[p]i,jII1,2,p0ijij為。jX]n,ijII．np()=PX；P()[p()nnnijmnmijppCKp()(nl)n()lijikkjkIPXjiX,pPXjXi(n)nmmPXiijmnmmi,Xk,XjPXmmlmnPXikTmPXi,Xk,XjPXi,XkmmlmnmmlPXi,XkPXikTmmlm(l)p(ml)p(m)pp(nl)(l)(nl)ikikkIkIP②P()nnnn4．X,nTpPXj0jnj0p(n)PXjnjPp,p,Tjn12P(n)p(np(nT12p(n)p(np()②(n)ppP(n)P(0)Pp()nTTnjijiIIPXi,Xi,,Xipppi1122nniii1in1nIGCDn:p0dd1d1(n)iif()injnijf．f()ijnn11f1iiifiiii()ii，niiin1i1ipip()。(n)n1fiiiin0n0j,ijjiiji與j互通，ii，ji，j1p0．狀態(tài)i(n)niCjC0iCIipijjC為閉集。如果CC,nTCXnX,nTnCjCp0(n)．CiijCCC中，它將永遠(yuǎn)留在C1ipiiiI,C,C,C集DCC12nnnf1DijD。C,C,CC。C12nDDCC，nnCnIDCCC，DC,C,C12n12np(n)ijp(n)ijlimpPXjXip(n)當(dāng)nij()nn0ijni,jIilim0pp(n)ijjnp，njjpp(n)jijlimp()nijn1limp(n)=,njI。ijj3．X,nIjI,pnijjpjiI1jI1,jI布。j0p()kkkij[Xt)]E[Xtt,t)EXt)Xt)R)R。XXXt),tT和t),tTEXtYt)及X()()()()EYtXttXt和Ytt,t)EXtYt)R)Rt,t)EYt)Xt)R)R(0)0R)R)R()RXXXXR)R(0)X是周期的，則相關(guān)函數(shù)Rt))limR()mmXXXt)Xt)Xt)與果X。XXX||t)Yt)和()()，R；聯(lián)合平穩(wěn)過(guò)程X的互相關(guān)函數(shù)，RXYRRRRXYYXXYR()R)()()。Xt和YtR()R()。XYYXXYYX略1t)l.mXt)dtTXTTT1t)Xt)l.mXt)XtdtTXTTTt)[Xtmt)XXXt)Xt)[Xt)Xt)]R)X[Xt[X(t)X(t)]t與E12T2T(1)(R)m02TTXXT12T2T(1)[B)R)0)[()()()()]2BEXtXtXtXt1TT1X111TXttTt),tXt)X(t)XtTTTXteXte(,T)FTjtjtTT得1T2tdtXtdtFTd,)X22TT對(duì)時(shí)間區(qū)間[T,T]lim1lim111lim1EXt)EF,T)d,)EFTdT222TTTTTTT1limT為Xt),tEXt)22TTT1(),為Xts(X)limEFT,)t2TTt),tXlim1lim1TTTTEXt)EXt)EXt)R2222TXTTT1d)s2X1RR)s)ed))ssedjjXXXX,n0,X對(duì)n1R(n))()Rne()sedjnjnXXXXn)R)s)XRes是的。jXXXXt),tR)R()s)，XXX1s(X)2R)dR)sd)cos()XXX0)②s是X)()XtXt()sX)sX2a2t)11)ea0aa222cos()))]sin()))]000000sinR(es))RTs))ejT0等j00XX0XX0t),tXXt),t()NsX0)N)ttX(t)X(t)和，和RX012121R))sed)ed)RRsjj1sed))edsRjs)s)s()s()XYXYXYYXs)s))s()()；若Xt和YtR)0s)s)0YX2。XYXYXY()H(j)H的。1H(hte)()ht())Hedjtjtt)Yt)X)R)h()R)h)h()RYXYXR)RXY))hXt()X性系統(tǒng)的未知脈沖響應(yīng)。因?yàn)镽)N)，X0Nu)huNh)R)R)h)X00R)h)N0s)H)s)2YXH)))HH2)s)H)s)R)反R2XYXY)R)h)s)H)s)s(Hs))R，XYXXYXYXX==T1服務(wù)時(shí)間符合負(fù)指數(shù)分布時(shí)，設(shè)它的概