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