随机过程(Poisson过程).pdf

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1、13ÙPoissonL§˜!Poisson!Poisson6n!àgPoissonL§o!^‡©ÙÊ!PoissonL§Ü¿†©)8!EÜPoissonL§Ô!šàgPoissonL§13ÙPoissonL§˜!PoissonPoisson<Ô013ÙPoissonL§Poisson©Ù13ÙPoissonL§!Poisson6Poisson6^u£ã‘ÅÑÖXÚ§X>&Û>{oÅÉ>{
¶•rc4SA?n𕶕cŒ.‡½ïû¬¶¦•cÿ•Ëÿú
ð•ÑÖXÚÚ–´½§¯kSüÐ

2、"‘Å5̇u•6§6þŒ´‘Å"Ï3u‡•3,ž•´ÄI‡ÑÖ´‘Å"13ÙPoissonL§Poisson6äkeá5µ(1)²•5c,ÑÖ–ϦÑÖ•6´²•¶(2)Õá5ˆ‡•3,ž•´Äc,ÑÖ–ϦÑÖƒpÕá¶(3)Dk5ážmS§•õ•k˜¶•Ï¦ÑÖ"13ÙPoissonL§þãá5êÆ)ºXeµ-N(0)=0"t>0ž§^N(t)L«(0;t]žmS,ÑÖXÚÉ•ÑÖgê"Ïd§N(t);t0´ëYžm!šKꊑÅL§§½¡ŠOêL§"

3、(1)²•5dN(t)N(s)=N(ts);s0ž§^N(t)L«rÝ•Poisson63(0;t]žmS•ê"@oN(

4、t)Ñlëê•tPoisson©Ùµ(t)ketP(N(t)=k)=;k=0;1;2;


;k!13ÙPoissonL§y²:(•{1)½t>0"é?¿n1§ò[0;t]©¡n‡«m§0=t0

5、=0)ntt=1P(N((0;])=1)+P(N((0;])2)nnDk5t1=1+o()nnÏd§t1nP(N(t)=0)=1+o()nnt=e;n!113ÙPoissonL§(2)k=1P(N(t)=1)Xn=PfN((tk1;tk])=1g;fN((ti1;ti])=0gk=1i6=kXnY=P(N((tk1;tk])=1)P(N((ti1;ti])=0)k=1i6=kttn1=nP(N((0;])=1)P(N((0;])=0)nnt1t1n1=n

6、(+o())1+o()nnnnt=te;n!113ÙPoissonL§(3)k=2P(N(t)=2)Xn=PfN((tk1;tk])=2g;fN((ti1;ti])=0gk=1i6=kX+PN((tk1;tk])=1;N((tj1;tj])=1;1k

7、(tk1;tk])=0gk=1i6=k!0;n!1XPN((tk1;tk])=1;N((tj1;tj])=1;1k2(t)ktP(N(t)=k)=ek!13ÙPoissonL§y²:(•{2)-pk(t)=P(N(t)=k)"(1)k=0p0(t+
t)=P(N(t+
t)=0)=P(N(t)=0;N((t;t+
t

8、])=0)=P(N(t)=0)P(N((t;t+
t])=0)=po(t)(1
t+o(
t))p0(t+
t)p0(t)=p0(t)
t++o(
t))13ÙPoissonL§0p0(t)=p0(t)5¿µN(0)=0§tp0(t)=e13ÙPoissonL§(2)k1pk(t+
t)=P(N(t+
t)=k)=P(N(t)=k;N((t;t+
t])=0)+P(N(t)=k1;N((t;t+
t])=1)+P(N