第3章-Poisson过程.pdf

1/47kJIkJIGoBackFullScreenCloseQuitAAALLL?IIIOOO?2/47kJIkJIGoBackFullScreenCloseQuit1113PoissonLLL 3.1 PoissonL 3.2 PoissonLX?eZ 3.3 PoissonL?23/47kJIkJIGoBackFullScreenCloseQuit3.1PoissonLLL?3.1.1 LN(t),t 0 OOOLLLXJN(t)Ll0?t,AAu)?g?eA:(1)N(t)0?;(2)s 0)?PoissonLXJ(1)N(0)=0;(2)LkO;4/47kJIkJIGoBackFullScreenCloseQuit(3)?s,t 0,P(N(t+s)N(s)=n)=et(t)nn!,n=0,1,2,555(1)PoissonL-O?OL(2)duEN(t)=t,u mSu)?g?PoissonL?r.3.1.1(PoissonL3?A)3Xy?Poisson L?.X,?o?8?,?PoissonL5,?l8:00md?Y10/?Kl9:00?10:001?Sk55d?V?l10:00-11:00vk 0,?h 0P(N(t+h)N(t)=1)=h+o(h);(4)0?h 0P(N(t+h)N(t)2)=o(h).KN(t),t 0PoissonL.r0,tyn?mmKd(4)0?n 3z?mSu)8/47kJIkJIGoBackFullScreenCloseQuitgg?Vu0du)g?Vp tn(w,p?)u)?V 1p 1tnTgBernoulli?.u)g=?u)=”2d(2)0?-O5N(t)?ungBernoulli?ogdPoisson?%CN(t)lt?Poisson.PoissonL?d?y:nnn 3.1.1 v(1)0(4)0?OLN(t),t 0PoissonLL5Poisson Lv4.y?OLN(t),t 0v(1)0(4)0,yyPoissonL.w?,I?yN(t)l9/47kJIkJIGoBackFullScreenCloseQuitt?Poisson=.PPn(t)=P(N(t)=n),n=0,1,2P(h)=P(N(h)1)=P1(h)+P2(h)+=1 P0(h)P0(t+h)=P(N(t+h)=0)=P(N(t+h)N(t)=0,N(t)=0)=P(N(t)=0)P(N(t+h)N(t)=0)=P0(t)P0(h)=P0(t)(1 h+o(h)(3)0,(4)0.10/47kJIkJIGoBackFullScreenCloseQuitdP0(t+h)P0(t)h=P0(t)+o(h)h,-h 0?P00(t)=P0(t).)d,?P0(t)=Ket,K.dP0(0)=P(N(0)=0)=1?K=1?P0(t)=et.11/47kJIkJIGoBackFullScreenCloseQuitn?n 1kPn(t+h)=P(N(t+h)=n)=P(N(t)=n,N(t+h)N(t)=0)+P(N(t)=n 1,N(t+h)N(t)=1)+P(N(t+h)=n,N(t+h)N(t)2)=Pn(t)P0(h)+Pn1(t)P1(h)+o(h)=(1 h)Pn(t)+hPn1(t)+o(h).uPn(t+h)Pn(t)h=Pn(t)+Pn1(t)+o(h),-h 0?P0n(t)=Pn(t)+Pn1(t)12/47kJIkJIGoBackFullScreenCloseQuit|8B)?Pn(t)=et(t)nn!=P(N(t)=n).L5yPoissonLv(1)0(4)0I?y(3)0,(4)0.13/47kJIkJIGoBackFullScreenCloseQuitd1n?P(N(t+h)N(t)=1)=P(N(h)N(0)=1)=ehh1!=hXn=0(h)nn!=h1 h+o(h)=h+o(h)P(N(t+h)N(t)2)=P(N(h)N(0)2)=Xn=2eh(h)nn!=o(h).3.1.3 A?u)/r?PoissonLN(t),t 0.XJzgu)VpU?Pe5M(t)L?t?Pe5?o,K14/47kJIkJIGoBackFullScreenCloseQuitM(t),t 0rp?PoissonL.duzgu)?PP?U?P?u)lPoisson.M(t)k-O?I?yM(t)lpt?Poisson.=t 0,kP(M(t)=m)=(pt)mm!ept.15/47kJIkJIGoBackFullScreenCloseQuitduP(M(t)=m)=Xn=0P(M(t)=m|N(t)=m+n)P(N(t)=m+n)=Xn=0Cmm+npm(1 p)n(t)m+n(m+n)!et=etXn=0(pt)m(1 p)t)nm!n!=et(pt)mm!Xn=0(1 p)t)nn!=et(pt)mm!e(1p)t=ept(pt)mm!.3.1.4 ez?lPoissonr16/47kJIkJIGoBackFullScreenCloseQuit?zC?VpC*dmvkXz?k?V.)d=?lrp?Poisson?V(pt)kk!ept 3.1.5*?LU(NlPoissonV pV?*?N.ez(k)3?VpK3NV?mk)3?(lpV?Poisson.17/47kJIkJIGoBackFullScreenCloseQuit3.2PoissonLLLXXX?eeeZZZkPoissonL?kPX3-1PoissonLN(t),t 0?a?1?F.Tn,n=1,2,3,L1ngu)?5T0=18/47kJIkJIGoBackFullScreenCloseQuit0.Xn,n=1,2,L1ng1n1gu)?mm.3.2.1XnTn?nnn 3.2.1 Xn,n=1,2,l?p.ykX1?5?X1 t?duN(t)=0,=(0,tSvku).dP(X1 t)=P(N(t)=0)=et,l?P(X1 t)=1 et.19/47kJIkJIGoBackFullScreenCloseQuit25wX2P(X2 t|X1=s)=P(N(s+t)N(s)=0|X1=s)=P(N(s+t)N(s)=0)(O5)=et.X2X1,l?.-E?n(.555 3.2.1 n3.2.1?(JAT?duPoissonLk-OL3?“-#m0T“P0?Ny?“P50A?.nnn 3.2.2 Tn,n=1,2,3ln?.20/47kJIkJIGoBackFullScreenCloseQuityduTn=Pni=1Xi,?dn?Xip?k?A/n=1d5?Tnln?3p,?.5?N(t)nTn t=1ngu)3tc?u?tu)?8?n.dP(Tn t)=P(N(t)n)=Xj=net(t)jj!.21/47kJIkJIGoBackFullScreenCloseQuit1?Tn?f(t)=Xj=net(t)jj!+Xj=net(t)j1(j 1)!=et(t)n1(n 1)!=n(n)tn1et.PoissonLq:3.2.1 OLN(t),t 0?PoissonL,XJzgu)?mm X1,X2,pl?.3.2.1J?PoissonL?1O?22/47kJIkJIGoBackFullScreenCloseQuitBI?)n?Xi,i=1,2,=?PoissonL?.3.2.1?l8:00mk?k?z?m?l20?K?12:00k?l?k9?V?)d?l?N(t)r3?PoissonLp?.?8:00KP(N(4)N(0)=n)=e12(12)nn!12=?12:00l?12.?k9?L?VP(N(4)=9)=e12(12)99!.23/47kJIkJIGoBackFullScreenCloseQuit 3.2.2 b,U?*?6(6PoissonL?Oz?*?36(.38:?12:mTU?vk*?6(?V.)?80N(t)L0?t*?6(KN(t)r3?PoissonLKkN(4)N(0)P(3 4)?38:?12:mTU?vk*?6(?VPN(4)N(0)=0=e123.2.2uuu)?nnn 3.2.3 30,tSAu)g?cJeAu)?30,t!.24/47kJIkJIGoBackFullScreenCloseQuitun=1?/us tP(T1 s|N(t)=1)=P(T1 s,N(t)=1)P(N(t)=1)=P(Au)3sc(s,tSAvku)P(N(t)=1)=P(N(s)=1)P(N(t)N(s)=0)P(N(t)=1)=ses e(ts)tet=stnnn 3.2.4 3N(t)=n?eu)?nT1,T2,Tn?f(t1,t2,tn)=n!tn,0 t1 t2 tn.25/47kJIkJIGoBackFullScreenCloseQuity?0 t1 t2 tn tn+1=t.?hi?ti+hi ti+1,i=1,2,nP(ti Ti ti+hi,i=1,2,n|N(t)=n)=P(N(ti+hi)N(ti)=1,N(ti+1)N(ti+hi)=0,1in,N(t1)=0)P(N(t)=n)=h1eh1hnehne(th1h2hn)et(t)n/n!=n!tnh1hn.?UN(t)=n(T1,Tn)?nf(t1,tn)=limhi01inP(tiTiti+hi,1in|N(t)=n)h1h2hn=n!tn,0 t1 t2 0(t 0)?gtLXJ(1)N(0)=0;(2)LkO;(3)P(N(t+h)N(t)=1)=(t)h+o(h);(4)P(N(t+h)N(t)2)=o(h).aquPoissonLgPoissonLkXe?d.3.3.2 OLN(t),t 0r(t)0(t 0)?g PoissonL,e(1)N(0)=0;(2)LkO;(3)?t 0,s 0,N(t+s)N(t)34/47kJIkJIGoBackFullScreenCloseQuitm(t+s)m(t)=Rt+st(u)du?t.5m(t)=Rt0(s)ds.tLgtLm=X:nnn 3.3.1?N(t),t 0r(t)?gPoissonL.?t 0-N(t)=N(m1(t)KN(t)r 1?tL.yyy kd(t)0 m(t)=Rt0(s)ds 0NO m1(t)3NO.?IyN(t),t 0v3.1!?(1)(4)(1),(2)JdN(t)?A5U?.eyv(3),(4).Pv(t)=m1(t)KN(t)=N(m1(t)=N(v(t).35/47kJIkJIGoBackFullScreenCloseQuit?v=m1(t),v+h0=m1(t+h),Kdh=m(v+h0)m(v)=Zv+h0v(s)ds=(v)h0+o(h0)?limh0+P(N(t+h)N(t)=1)h=limh00+P(N(v+h0)N(v)=1)(v)h0+o(h0)=limh00+(v)h0+o(h0)(v)h0+o(h0)=1=P(N(t+h)N(t)=1)=h+o(h).36/47kJIkJIGoBackFullScreenCloseQuitn?P(N(t+h)N(t)2)=o(h).N(t),t 01?PoissonL.555:dnzgPoissonL?K?PoissonL?1?.,?1?=l?PoissonL?Er(t)?gPoissonL.3.3.1?,?10c3c5cS37/47kJIkJIGoBackFullScreenCloseQuit2.5I?g?5c2cI?g3S?Lg?V.)gPoissonLr(t)=(12.5,0 t 512,5 t 10m(10)=Z100(t)dt=Z5012.5dt+Z10512dt=4.5dP(N(10)N(0)=1)=e4.5(4.5)11!=92e92.3.3.2EEEPoissonLLL 3.3.3 LX(t),t 0EtL38/47kJIkJIGoBackFullScreenCloseQuit,XJut 0X(t)LX(t)=N(t)Xi=1YiN(t),t 0PoissonLYi,i=1,2,x?CN(t),t 0?.555EPoissonLOL?Yic,i=1,2,c,ztL.3.3.2?xi?gltLN(t)zg?G?7?YipkF,zg?u)?K0,tm39/47kJIkJIGoBackFullScreenCloseQuitmS?xiI?G?o7?X(t)EtLX(t)=N(t)Xi=1Yi 3.3.3?1?X?,X?mS1,S2,/r?PoissonL3zSn,n=1,2,k?.YnL3Sn?bYn,n=1,2,pSnK30,tmmS?X?oEPoissonL5.3.3.4 b?U?PoissonL?Aqb?7?/x?CX(t)P?mt?3dAs?40/47kJIkJIGoBackFullScreenCloseQuito?X(t),t 0EPoissonLnnn 3.3.2?X(t)=PN(t)i=1Yi,t 0EtLtLN(t),t 0?r,K(1)X(t)kO;(2)eE(Y2i)+,KEX(t)=t EY1,varX(t)=t E(Y21).yyy(1)-0 t0 t1 t2 tnKX(tk)X(tk1)=N(tk)Xi=N(tk1)+1Yi,k=1,2,ndLL?O59Yi,(i=1,2,n)m?5J?X(t)?O5.41/47kJIkJIGoBackFullScreenCloseQuit2|1kkt(u)=E(euXt)=Pn=0EeuXt|N(t)=nP(N(t)=n)=Pn=0Eeu(Y1+Yn)|N(t)=net(t)nn!=Pn=0Eeu(Y1+Yn)et(t)nn!=Pn=0E(euY1)E(euYn)et(t)nn!=Pn=0E(euY1)net(t)nn!=etE(euY1)1.?EX(t)=tEY19var(X(t)=tE(Y21).3.3.5 3?x?.?xi?rz?g?tLzg?Gl42/47kJIkJIGoBackFullScreenCloseQuit10000?Kc?xi?G?.):dn 3.3.2?EX(12)=2 12 10000=240000(?)3.3.6?z6?,|LPoissonL5.q?T|?z?V0.9z?pK?T|?U(12?)3T|?.)N1(t)L3m(0,tS?T|?KN1(t),t 0=6(/)?PoissonL.2N2(t)L3m(0,tS3T|?43/47kJIkJIGoBackFullScreenCloseQuit?Yi=(1,XJ1i?3T|0,XJ1i?3T|KYiuB(1,0.9)N1(t),t 0N2(t)=N1(t)Xi=1YiN2(t)P(5.4t)=N2(t),t 0=5.4(/)?PoissonLt=12?=720KN2(720)P(3888)=U(12?)3T|?EN2(720)=3888.44/47kJIkJIGoBackFullScreenCloseQuit555eZiL?T|?1i?3T|s?a(:?)kZi B(200,0.5)KN3(t)=N1(t)Xi=1Zi3m(0,tST|?E?KT|U?E?EN3=EZ1EN1=(2000.5)(61260)=432000?3.3.3PoissonLLLPoissonL?kX“x0?Nu),?XJoN?N3?Xu)?5 03=?eOLN(t),t 0?tL.KN(t),t 0tL.555?GoJN3t?mmSu)ng?VP(N(t+s)N(s)=n)=Z0et(t)nn!dG()?V.nnn 3.3.3?N(t),t 0tLE(2),K1EN(t)=tE;2varN(t)=t2var()+tE.46/47kJIkJIGoBackFullScreenCloseQuityyy(1)EN(t)=EE(N(t)|)=E(t)=tE;(2)var(N(t)=EN2(t)EN(t)2=EE(N2(t)|)(tE)2=E(t)2+t t2(E)2=t2var()+tE.3.3.7?u),KkU1,2P(=1)=p,P(=2)=1 p=q,0 p 1.?tu)?ng?.eg?3t+sc?5?V.,?u)1?V?47/47kJIkJIGoBackFullScreenCloseQuit):JP(t,t+s)S?|N(t)=n)=P2i=1P(=i)P(N(t)=n,N(t+s)N(t)=0|=i)P2i=1P(=i)P(N(t)=n|=i)=p(1t)ne1(s+t)+(1 p)(2t)ne2(s+t)p(1t)ne1t+(1 p)(2t)ne2t=pn1e1(s+t)+qn2e2(s+t)pn1e1t+qn2e2t9P(=1|N(t)=n)=pe1t(1t)npe1t(1t)n+(1 p)e2t(2t)n.