The Poisson Process

Stat 6402

Counting process

A stochastic process {N(t) : t ≥ 0} is a counting process if N(t) represents the total number of “events” that occur by time t. eg, # of persons entering a store before time t, # of people who were born by

time t, # of goals a soccer player scores by time t.

N(t) should satisfy: N(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 occur in (s, t]

http://www.ms.uky.edu/~mai/java/stat/countpro.html

Independent and stationary increments characterization of the Poisson process

The counting process {N(t), t>0} is said to be a Poisson process having rate λ if

1. N (0) = 02. The process has independent increments.

A counting process is said to have independent increments if for any non-overlapping intervals of time, I1 and I2, N(I1) and N (I2) are independent.

3. The number of events in any interval of length t is Poisson with mean λt. A counting process is said to have stationary increments if for all t>0,

s>0, N(t+s)- N(s) has a distribution that only depends on t, the length of the time interval.

Here, P{N(t+s)- N(s)=n} = e−λt(λt)n/n!, n = 0, 1, … E[N(t)] = λt

Little o(t) results

Let o(t) denote any function of t that satisfies o(t)/t → 0 as t → 0. Examples include o(t) =tn, n>1

For a Poisson process,

P(N(t) = 1) = λt + o(t)

P(N(t) > 1) = o(t).

Interarrival and waiting time distribution

{Tn, n= 1, 2, …} is called the sequence of interarrival times. P{T1>t} = P{N(t)=0} = e-λt

P{T2>t | T1 = s} = P{0 events in (s, s+t] | T1 = s} = P{0 events in (s, s+t] } = e-λt

Tn are iid exponential with mean 1/λ

Sn is the arrival time of the nth event, also called the waiting time until the nth event. Sn = T1+ T2+…. +Tn Sn has a gamma distribution with parameters n and λ

Partitioning a Poisson random variable Suppose each event is classified as a type I event with

probability p or a type II event with probability 1-p. If N(t) Poiss(α) and if each object of X is, independently, ∼

type 1 or type 2 with probability p and q = 1− p, then in fact N1(t) Poiss(pα), N∼ 2(t) Poiss(qα) and they are ∼independent.

Suppose there are k possible types of events, represented by Ni(t), i=1, …k, then they are independent Poisson random variables having means

E[Ni(t)] = λt

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Conditional distribution of the arrival times

Theorem: Given that N(t)=n, the n arrival times S1, S2,…Sn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval (0, t).

f (S1,… Sn | n) = n!/ tn, 0<S1…<Sn

Proposition : Given that Sn=t, the set S1,…Sn-1 has

the distribution of a set of n-1 independent uniform (0,t) random variables.

Compound Poisson process

A stochastic process {X(t), t>0} is said to be a compound Poisson process if it can be represented as X(t) =

Here {N(t), t>0} is a Poisson process, and {Yi, i>0} is a family of independent and identically distributed random variables that is also independent of N(t).

eg. Suppose customers leave a supermarket with a Poisson process. Yi represents the amount spent by the ith customer and they are iid. Then {X(t), t>0} is a compound Poisson, X(t) denotes the total amount of money spent by time t.

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tN

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Yi

Compound Poisson process

E[x(t)] = λt E[Yi]

Var[x(t)]=λt E[Yi 2]

Suppose that families migrate to an area at a Poisson rate 2 per week. If the number of people in each family is independent and takes on the values 1, 2, 3, 4 with respective probabilities 1/6, 1/3, 1/3, 1/6, then what is the expected value and variance of the number of individuals migrating to this area during a fixed five-week period?