The Poisson Process 英文文献资料

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1、Chapter3：ThePoissonProcess3.1TheExponentialDistributionDefinition:Acontinuousrandomvariable?issaidtohaveexponentialdistributionwithparameter??>0,ifithasp.d.f.??−??,?≥0??=0,?<0orequivalently,withc.d.f.?1−?−??,?≥0??=????=0,?<0−∞PropertiesofExponentialDistributionTh

2、eorem.Exponentialdistributionwiththerate?=1Exponentialdistributionwiththemeanof?Proof:∞??=???−????0∞=?−?−??−∞−?−????\Integrationbypart00∞=0−0−?−??/?01=?PropertiesofExponentialDistributionLet?1,?2,⋯,??bei.i.d.exponentialrandom?variableswithparameter?,then?=?=1??i

3、sgammadistributedwithparameter?and?:????=??−1?−??,?≥0.?Γ?Proof:Homework(hint:inductionon?)PropertiesofExponentialDistributionLet?1and?2beindependentexponentialrandomvariableswithparameter?1and?2,respectively.Then?1??1

4、?2=???2???\con

5、ditionon?2∞−?2?=0??1?=??1

6、>?,?2>?,⋯,??>??=?=1???>?=??−????=1−(??=??=1?)?MemorylessPropertyofExponentialDistributionTheorem.AnexponentiallydistributedrandomvariableXismemoryless.Moreprecisely,forallt0andx0,??>?+?

7、?>?=??>?orEquivalently,??>?+?=??>???>?//Havingwaitedfortunittimes,theresidu

8、al//waitingtimestillhasthesamedistribution.//Or,不要问我等了多久Proof:??>?+??>??(?>?+?,?>?)=\Conditionalprobability?(?>?)?(?>?+?)=\?≥0?(?>?)?−?(?+?)=\Definitionofexp.randomvariables?−??=?−??=?(?>?)\Amemorylessdist.isnon-negative.Thediscreteanalogueisthegeometricdistr

9、ibution.Geometric(p)distribution:??>?+??>?=?(?>?)Homework:YouarewaitingforGodot,whowillarriveontheNthbus,whereNisGeometric(p)distributed.Iftheinter-arrivaltimesofbusesareExponential()i.i.d.,whatisthedistributionofyourwaitingtimeforGodot?UniquenessTheorem.Exponen

10、tialdistributionistheonlycontinuousmemorylessdistribution.Proof:LetXbeanonnegativecontinuousr.v.suchthat??>?+?

11、?>?=??>?forall?≥0and?≥0.Equivalently,??>?+?=??>?