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

2. 12.2 – Poisson Distribution
Definition: 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).
Expectation E[X] = µ = tλ and variance Var(X) = µ = tλ

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

4. 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):
Each arrival time Xi, is a random variable with Gam(i, λ) distribution for α=i :
We also observe that Gam(1, λ) = Exp(λ):

5. Example form Baron Book:

6. 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

7. 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 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