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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,不要問我等了多久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,??>?+?=??>?
