The Poisson Process

1. The Poisson Process Stat 6402

2. 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

3. 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. • 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

4. 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).

5. 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

6. 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), 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 variables having means • E[Ni(t)] = λ

7. 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.

8. 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.

9. 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?