CIS 2033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007

Instructor Longin Jan Latecki

C12: The Poisson process

12.2 – Poisson DistributionDefinition: A discrete RV X has a Poisson distribution with parameter µ, where µ > 0 if its probability mass function is given by

for k = 0,1,2…,

where µ is the expected number of rare events occurring in time interval [0, t], which is fixed for X. We can express µ = t λ, where t is the length of the interval, e.g., number of minutes. Hence

λ = µ / t = number of events per time unite = probability of success.

λ is also called the intensity or frequency of the Poisson process. We denote this distribution: Pois(µ) = Pois(tλ).

ek!

k)P(X P(k)k

Expectation E[X] = µ = tλ and variance Var(X) = µ = tλ

The inter-arrival times T1=X1, T2=X2 – X1, T3=X3 – X2 … are independent RVs, each

with an Exp(λ) distribution.

Hence expected inter-arrival time is E(Ti) =1/λ. Since for Poisson

λ = µ / t = (number of events) / (time unite) = probability of success,

we have for the exponential distribution

E(Ti) =1/λ = t / µ = (time unite) / (number of events) = wait time

Let X1, X2, … be arrival times such that the probability of k arrivals in a given time interval [0, t] has a Poisson distribution Pois(tλ):

The differences Ti = Xi – Xi-1 are called inter-arrival times or wait times.

tkt e

k!

)(k)P(X

Each arrival time Xi, is a random variable with Gam(i, λ) distribution for α=i :

Let X1, X2, … be arrival times such that the probability of k arrivals in a given time interval [0, t] has a Poisson distribution Pois(λt):

tkt e

k!

)(k)P(X

We also observe that Gam(1, λ) = Exp(λ):

Example form Baron Book:

12.2 –Random arrivals

Example: Telephone calls arrival times Calls arrive at random times, X1, X2, X3…

Homegeneity aka weak stationarity: is the rate lambda at which arrivals occur in constant over time: in a subinterval of length u the expectation of the number of telephone calls is λu.

Independence: The number of arrivals in disjoint time intervals are independent random variables.

N(I) = total number of calls in an interval I Nt=N([0,t])

E[Nt] = t λ Divide Interval [0,t] into n intervals, each of size t/n

12.2 –Random arrivals

When n is large enough, every interval Ij,n = ((j-1)t/n , jt/n] contains either 0 or 1 arrivals.Arrival: For such a large n ( n > λ t),

Rj = number of arrivals in the time interval Ij,n, Rj = 0 or 1

Rj has a Ber(p) distribution for some p.Recall: (For a Bernoulli random variable)

E[Rj] = 0 • (1 – p) + 1 • p = p

By Homogeneity assumption for each j p = λ • length of Ij,n = λ ( t / n)

Total number of calls: Nt = R1 + R2 + … + Rn.

By Independence assumptionRj are independent random variables, soNt has a Bin(n,p) distribution, with p = λ t/n

When n goes to infinity, Bin(n,p) converges to a Poisson distribution