Homogeneous Poisson Process

Question 1

Counting processes
▪ Definition

Answer 1

A stochastic process (N(t))ₜ˲₀
is said to be a counting process if N(t) represents the total number of certain random events that occur by time t (starting from time 0)

Question 2

Counting processes
▪ Requirements

Answer 2

A counting process (N(t))ₜ˲₀ must satisfy:
▪ N(0) = 0 and N(t)>0 for all t>0
▪ N(t) is integer valued
▪ if s<t then N(s)⩽N(t)
▪ for s<t, N(t) - N(s) equals the number of events that occu in the time interval (s,t]

Question 3

Counting processes
▪ Examples of counting processes

Answer 3

Counting processes have been used to model arrivals then we refer the occurrence of each event as an Arrival

Example
▪ N(t) = number of persons who enter a shop
▪ (N(t))ₜ˲₀ = counting process in which an event corresponds to a person entering the store

Question 4

Independent increments
▪ Definition

Answer 4

Let (N(t))ₜ˲₀ be a counting process it has independent increments if
for all
0⩽t₀<t₁<…<tₙ
the random variables
N(tₙ)-N(tₙ₋₁),…,N(t₂)-N(t₁),N(t₁)-N(t₀)
are independent

(Number of arrivals in disjoint time intervals are independent)

Question 5

Stationary increments
▪ Definition

Answer 5

Let (N(t))ₜ˲₀ be a counting process it has stationary increments if
for
0⩽t₁<t₂ and s>0
the random variables
▪ N(t₂₊ₛ)-N(t₁₊ₛ)
▪ N(t₂)-N(t₁)
have the same distribution

(The distribution increment depends only on the length of the considered time interval and not on the exact location)

In particular
N(t₂)-N(t₁) for all 0⩽t₁<t₂
has the same distribution as
N(t₂-t₁)

Question 6

Poisson process
▪ Definition ①

Answer 6

(Usually used in scenarios where we are counting the occurrences of certain events that appear to happen at certain rate but completely at random)

Let λ>0 be fixed.
A counting process (N(t))ₜ˲₀ is called a Poisson process with rate/intensity λ if:
▪ N(0) = 0
▪ (N(t))ₜ˲₀ has independent increments
▪ the number of arrivals in any interval of length ͳ>0 has Poisson distribution with parameter λͳ

(Number of arrivals depends only on the length of the interval and not on the exact location of the interval.
therefore the Poisson process has stationary increments)

Question 7

Poisson process
▪ Arrival and interarrival times

Answer 7

Consider a counting process (N(t))ₜ˲₀ with rate λ

Denote
s₁ = time of the first event
sₙ = elapsed time between the (n-1)st and the n-th event (for n>1)

The sequence (Sₙ)ₙϵN is called the sequence of the interarrival times

Question 8

Poisson process
▪ Distribution of the interarrival times

Answer 8

Let (N(t))ₜ˲₀ be a Poisson process with rate λ
Then the interarrival times
s₁, s₂, …, sₙ
are independent and identically distributed with
sᵢ~Exp(λ)

P(sᵢ>t)=P(N(t))=0)=e^-λt

Question 9

Poisson process
▪ Definition ②

Answer 9

Let λ>0 be fixed.
A counting process (N(t))ₜ˲₀ is called a Poisson process with rate/intensity λ if:
▪ N(0) = 0
▪ the sequence of interarrival times is a sequence of iid random variables with exponential distribution with parameter λ

Question 10

Poisson process
▪ Distribution of the arrival times

Answer 10

Let (N(t))ₜ˲₀ be a Poisson process with rate/intensity λ then:

Tₙ~Γ(n,λ) for any n ϵ N

In particular
E(Tₙ) = n/λ
Var(Tₙ) = n/λ²