MATH 3033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007

Slides by Gautam ShankarFormat by Tim Birbeck

Instructor Longin Jan Latecki

C12: The Poisson process

12.1 – Random Points Poisson process model: often applies in situations where

there is a very large population, and each member of the population has a very small probability to produce a point of the process.

Examples of Random points: arrival times of email messages at a server, the times at which asteroids hit earth, arrival times of radioactive particles at a Geiger counter, times at which your computer crashes, the times at which electronic components fail, and arrival times of people at a pump in an oasis.

12.2 – Taking a closer look at 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 lambda * u.

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

N(I) = total number of calls in an interval I N([0,t]) Nt E[Nt] = λ t Divide Interval [0,t] into n intervals, each of size t/n

12.2 – Taking a closer look at random arrivals When n is large enough, every interval Ij,n = ((j-1)t/n , jt/n] will contain

either 0 or 1 arrival.Arrival: For such a large n ( n > λ t),

Rj = number of arrivals in the time interval Ij,n Rj has a Ber(pj) distribution for some pj.

Recall: (For a Bernoulli random variable) E[Rj] = 0 • (1 – pj) + 1 • pj = pj

By Homogeneity assumption (see prev slide), for each j pj = λ • length of Ij,n = ( λ t / n)

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

By Independence assumption (see prev slide)Rj are independent random variables, soNt has a Bin(n,p) distribution, with p = λ t/n

12.2 – Taking a closer look at random arrivals

Definition: A discrete random variable X has a Poisson distribution with parameter µ, where µ > 0 if its probability mass function p is given by

for k = 0,1,2..

We denote this distribution by Pois(µ)

ek!

k)P(X P(k)k

The Expectation an variance of a Poisson DistributionLet X have a Poisson distribution with parameter µ; then

E[X] = µ and Var(X) = µ

12.3 – The one-dimensional Poisson process

Interarrival TimesThe differences

Ti = Xi – Xi-1 are called interarrival times.This imples that

t-t11 e - 1 0) P(N - 1 t) P(T - 1 t) P( T

Therefore T1 has an exponential distribution with parameter λ

t-

112

e 0) t]) sP(N((s,

t])s (s,in arrivals P(no

s) T | t] s (s,in arrivals P(no s) T | t P(

T

12.3 – The one-dimensional Poisson process

T1 and T2 are independent, and

t-2 e t)P( T

The one-dimensional Poisson process with intensity λ is a sequence , , ,.. Of random variables having the property that the inter-arrival times are independent random variables, each with an Exp(λ) distribution.

is equal to the number of Xi that are smaller than(or equal to) t.

1X,...,, 23121 XXXXX

2X 3X

tN