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C12: The Poisson process

CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Instructor Longin Jan Latecki. C12: The Poisson process. 12.2 –Random arrivals. Example: Telephone calls arrival times Calls arrive at random times, X 1 , X 2 , X 3 …

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C12: The Poisson process

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  1. CIS 2033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007 Instructor Longin Jan Latecki C12: The Poisson process

  2. 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] = λ t • Divide Interval [0,t] into n intervals, each of size t/n

  3. 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 jp = λ• length of Ij,n = λ (t / n) • Total number of calls:Nt = R1 + R2 + … + Rn. • By Independence assumption Rj are independent random variables, so Nt has a Bin(n,p) distribution, with p = λ t/n • When n goes to infinity, Bin(n,p) converges to a Poisson distribution

  4. 12.2 – Poisson Distribution 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.. where µ=λt, and t is the length of the interval for which we consider X. λ is called the intensity or frequency of the Poisson process. We denote this distribution by Pois(µ) =Pois(λt). Expectation an variance of a Poisson distribution: E[X] = µ = λt and Var(X) = µ = λt

  5. 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): The differences Ti = Xi – Xi-1 are called inter-arrival times. The inter-arrival times T1=X1, T2=X2 – X1, T3=X3 – X2 … are independent random variables, each with an Exp(λ) distribution.Hence expected inter-arrival timeis E(Ti) =1/λ. Each arrival time Xi, is a random variable with Gam(i, λ) distribution for α=i :

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