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1、Chapter2Poissonprocesses2.1Definition2.2PropertiesofPoissonprocesses2.3NonhomogeneousPoissonprocesses2.4CompoundPoissonprocesses2.5FilteredPoissonprocesses22.1DefinitionConsideracountingprocessN={N(t),t≥0},whereN(t)denotesthenumberofarrivalsintheinterval(0,t].Definition1Acount
2、ingprocessN={N(t),t≥0}isaPoissonprocesswithrateλ>0,ifitpossessesthefollowingproperties:(a)N(0)=0,(b)Itsatisfiesthestationaryandindependentincrementproperties(c)P{N(h)=1}=λh+o(h)(d)P{N(h)≥2}=o(h)32.1DefinitionDefinition2:Thecountingprocess{N(t),t≥0}issaidtobeaPoissonprocesshavi
3、ngarrivalrateλ,λ>0,if(a)N(0)=0(b)Theprocesshasindependentincrements(c)ThenumberofeventsinanyintervaloflengthtisPoissondistributedwithmeanλt.t≥0n−t(λt)P{N(t)=n}=eλ,n=0,1,2,......(Eq.2-1-1)n!E[N(t)]=λt42.2PropertiesofPoissonprocesses¢InterarrivaltimedistributionLetSdenotestheepo
4、chofthentharrivalofNandndefineS=0.TheintervaltimeX=S-S,so0nnn-1nSn=∑Xk,n=1,2......k=1Proposition(命题）2.1:APoissonprocessN={N(t),t≥0}withrateλ,theinterarrivaltime{X,n≥1}areindependentlyandidenticallydistributednrandomvariables,eachfollowinganexponentialdistributionwithparameterλ
5、.(f(t)=-λtλe,t>0,mean=1/λ)52.2PropertiesofPoissonprocessesProof:{X>t}⇔{N(t)=0}1P{X1>t}=P{N(t)=0}=e-λtP{X≤t}=1-e-λt1differentiatingthetwosidesofequationwithrespecttot,WeobtainthedensityofdistributionofX:1−λtf(t)=λeXfollowsanexponentialdistributionwithparameterλ162.2Propertiesof
6、PoissonprocessesForanys>0andt>0,{X>t
7、X=s}⇔{0eventin(s,s+t]
8、X=s}211P{X>t
9、X=s}=P{0eventin(s,s+t]
10、X=s}211=P{0eventin(s,s+t]}(independent-increment)=P{0eventin(0,t]}(stationary-increment)=P{N(t)=0}=e-λtP{X≤t
11、X=s}=1-e-λt21−λtf(t)=λeXfollowsanexponentialdistributionwithparameterλ,2a
12、ndXandXareindependent1272.2PropertiesofPoissonprocessesSimilarly,weobtain:P{X>t
13、X=s}=e-λtnn-1P{X≤t
14、X=s}=1-e-λtnn-1Weobtaintheinterarrivaltimedistribution:−λtf(t)=λeeachinterarrivaltime{X,n≥1}followsanexponentialndistributionwithparameterλ.82.2PropertiesofPoissonprocessesExampl
15、e1SupposethatpeopleimmigrateintoaterritoryataPoissonrateλ=1pe
