Post on 30-Oct-2021
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Second definition of the Poisson process
Preliminary Cor recall definition of o h
Definition A function f is och if nymof o
Example fch hate och if E O h
e 16ththat X
few eh I ochh
e tht k t 40 as h o
far heh 1 och
Remain oChl toChl ochcoath Ch t constant
Definition Poissonprocess A counting process
Nlt to is a
poisonprocess of rated ofit satisfies the following
axioms
N o D
it has independentincrements
PCNitth Nlt De th t oCh
4 PCNHth NIH 22 och
or equivalently PCN Ltth Nlt o l th och
Remain A consequence of this 2nd definition is that
N tts NCS Poisson dt
in particular we have the property of stationaryincrements
Proofof Remark We have that Bin K Ih E Poisson d forlarger
let us divide Sitts into u increments
InIn I2 I i e i l l l l l letsS tk
P some Ij has 22 events GI PCI has 22 events
El IE octa k oCE t.no I o as ked
const k
Nlt intervals where 1 event occurs as a
P Ij has 1 event that OctaIN It e independentlypricing
e of k intervalsbyG each w p Hq to Cta
Nlt n Binomial CK p diet Octa
2 Poisson dt tko PoissonCdt
0 as re x
we will use this to show the 2definitions are equivalent
theorem Nlt is a Poisson process of rate A 2nd definition
NUT is a country process withi i d intervals EapQ
est definition
roof We already saw that indep increments holdsholds by def
P Nlt th Nlt 1 PCN h l them the Ah
4h11 t Hh 7 t Ah t och
C P NHth Nlt 22 r PCN h L PINCH 0
toth th t oCh Il th och
h dh to Ch th e th
1 0 h XtXht ochoch
J we need to check that interamual times T Tz Tz
are i i d Exp A rub
o_OI ttf
Recall that we showed Nttts Nlt n Poisson Ks
Distributonofti PLT stl p Nlt o EdtIn Expat
Distributional PCTzst fPast IT 9ft cods
conditioning on Ts
PCT t T s PCN Stt Nlt o sett
IT is independent of Ti
PCTz t J eHfi Is e
t
Tz EepCd
This argument can be extended to PCTunst Tat ten S
byinduction showing that Taitz are i id EapCdl
Z.FurtherpropertiesTheorem Superposition let Nlt t 20 I Ndt too be
independent Poisson processes of rates dn.dz let N E EN Nd
Then Nlt is a Poisson process of rate ditka
Proof We use the second definition
N O N LO t Nz O
y independent increments s Sz et t
N Csu Nics indep of N Ltd N Ct
Neese Nds indef of Nz Ctu Mach
N Sz Ncs indep of NLtz Nlt
Pl N Ith Nlt L PCNftth N IE L Nectth Nut 0
PCN Ctu N 14 0 Nzltth Notts 1
video ofN and NzPCN 4th NIH 1 PCNectth Notts
pcNaltth N CH o PCNzLtth Nut L
Cd htoch e dah to h tcdzhto.ch t d.htoChD
Xih d.dzh2toCh toCh1tdzh ddzh2toCh
datdat h t och
4 Similarly show exercise that PCNUth Nlt o
Chtdath och