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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.
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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:

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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) = 0
2. 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.
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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.
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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.
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Sn = T1+ T2+…. +Tn
Sn has a gamma distribution with parameters n and λ
Partitioning a Poisson random variable
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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), N2(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
t
variables having
means
E[Ni(t)] = λ
 Pi(s)ds
0
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) = N (t )
Yi
i 1

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.
Compound Poisson process
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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?