The Poisson Process Presented by Darrin Gershman and Dave Wilkerson

Overview of Presentation Who was Poisson? What is a counting process? What is a Poisson process? What useful tools develop from the Poisson process? What types of Poisson processes are there? What are some applications of the Poisson process?

Siméon Denis Poisson Born: 6/21/1781- Pithiviers, France Died: 4/25/1840- Sceaux, France “Life is good for only two things: discovering mathematics and teaching mathematics.”

Siméon Denis Poisson Poisson’s father originally wanted him to become a doctor. After a brief apprenticeship with an uncle, Poisson realized he did not want to be a doctor. After the French Revolution, more opportunities became available for Poisson, whose family was not part of the nobility. Poisson went to the École Centrale and later the École Polytechnique in Paris, where he excelled in mathematics, despite having much less formal education than his peers.

Poisson’s education and work Poisson impressed his teachers Laplace and Lagrange with his abilities. Unfortunately, the École Polytechnique specialized in geometry, and Poisson could not draw diagrams well. However, his final paper on the theory of equations was so good he was allowed to graduate without taking the final examination. After graduating, Poisson received his first teaching position at the École Polytechnique in Paris, which rarely happened. Poisson did most of his work on ordinary and partial differential equations. He also worked on problems involving physical topics, such as pendulums and sound.

Poisson’s accomplishments Poisson held a professorship at the École Polytechnique, was an astronomer at the Bureau des Longitudes, was named chair of the Faculté des Sciences, and was an examiner at the École Militaire. He has many mathematical and scientific tools named for him, including Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity. He first published his Poisson distribution in 1837 in Recherches sur la probabilité des jugements en matière criminelle et matière civile. Although this was important to probability and random processes, other French mathematicians did not see his work as significant. His accomplishments were more accepted outside France, such as in Russia, where Chebychev used Poisson’s results to develop his own.

Counting Processes {N(t), t  0} is a counting process if N(t) is the total number of events that occur by time t Ex. (1) number of cars passing by, EX. (2) number of home runs hit by a baseball player Facts about counting process N(t): (a) N(t)  0 (b) N(t) is integer-valued for all t (c) If t > s, then N(t)  N(s) (d) If t > s, then N(t)-N(s)=the number of events in the interval (s,t]

Independent and stationary increments A counting process N(t) has: independent increments: if the number of events occurring in disjoint time intervals are independent. stationary increments The number of events occurring in interval (s, s+t) has the same distribution for all s (i.e., the number of events occurring in an interval depends only on the length of the interval). Ex. The Store example

Poisson Processes Definition 1: Counting process {N(t), t  0} is a Poisson process with rate, > 0, if: (i) N(0)=0 (ii) N(t) has independent increments (iii) the number of events in any interval of length t ~ Poi( t) (  s,t  0, P{N(t+s) –N(s) = n} = From condition (iii), we know that N(t) also has stationary increments and E[N(t)]= t Conditions (i) and (ii) are usually easy to show, but condition (iii) is more difficult to show. Thus, an alternate set of conditions is useful for showing some N(t) is a Poisson process.

Alternate definition of Poisson process {N(t), t  0} is a Poisson process with rate, > 0, if: (i) N(0)=0 (ii) N(t) has stationary and independent increments (iii) P{N(h) = 1} = h + o(h) (iv) P{N(h)  2} = o(h) where function f is said to be o(h) if The first definition is useful when given that a sequence is a Poisson process. This alternate definition is useful when showing that a given object is a Poisson process.

Theorem: the alternate definition implies definition 1. Proof: Fix, and let by independent increments by stationary increments Assumptions (iii) and (iv) imply

Conditioning on whether N(h) = 0, N(h) = 1, or N(h) 2 implies As we get, Which is the same as

Integrating and setting g(0)=1 gives, Solving for g(t) we obtain, This is the Laplace transform of a Poisson random variable with mean.

Interarrival times We will now look at the distribution of the times between events in a Poisson process. T 1 = time of first event in the Poisson process T 2 = time between 1 st and 2 nd events T n = time between (n-1)st and nth events. {T n, n=1,2,…} is the sequence of interarrival times What is the distribution of T n ?

Distribution of T n First consider T 1 : P{T 1 > t} = P{N(t)=0} = e - t (condition (iii) with s=0, n=0) Thus, T 1 ~ exponential( ) Now consider T 2 : P{T 2 >t | T 1 =s} = P{0 events in (s,s+t] | T 1 =s} = = P{0 events in (s,s+t] (by stationary increments) = P{0 events in (0,t]} (by independent increments) = P{N(t)=0} = e - t Thus, T 2 ~ exponential( ) (same as T 1 ) Conclusion: The interarrival times T n, n=1,2, … are iid exponential( ) (mean 1/ ) Thus, we can say that the interarrival times are “ memory less. ”

Waiting Times We say S n, n=1,2,… is the waiting time (or arrival time) until the nth event occurs. S n =, n  1 S n is the sum of n iid exponential( ) random variables. Thus, S n ~ Gamma(n, 1/ )

Poisson processes with multiple types of events Let {N(t), t  0} be a Poisson process with rate Now partition events into type I, II p=P(event of type I occurs), 1-p=P(event of type II occurs) N 1 (t) and N 2 (t) are the number of type I and type II events Results: (1) N(t) = N 1 (t) + N 2 (t) (2) {N 1 (t), t  0} and {N 2 (t), t  0} are Poisson processes with rates p and (1-p) respectively. (3) {N 1 (t), t  0} and {N 2 (t), t  0} are independent. example: males/females Poisson processes that have more than 2 types of events yield results analogous to those above.

Nonhomogeneous Poisson Processes A nonhomogeneous Poisson process allows for the arrival rate to be a function of time (t) instead of a constant. The definition for such a process is: (i) N(0)=0 (ii) N(t) has independent increments (iii) P{N(t+h) – N(t) = 1} = (t)h + o(h) (iv) P{N(t+h) – N(t)  2} = o(h) Nonhomogeneous Poisson processes are useful when the rate of events varies. For example, when observing customers entering a restaurant, the numbers will be much greater during meal times than during off hours.

Compound Poisson Processes Let {N(t), t  0} be a Poisson process and let {Y i, i  1} be a family of iid random variables independent of the Poisson process. If we define X(t) =, t  0, then {X(t), t  0} is a compound Poisson process. ex. At a bus station, buses arrive according to a Poisson process, and the amounts of people arriving on each bus are independent and identically distributed. If X(t) represents the number of people who arrive at the station before time t.

Order Statistics If N(t) = n, then n events occurred in [0,t] Let S 1,…S n be the arrival times of those n events. Then the distribution of arrival times S 1,…S n is the same as the distribution of the order statistics of n iid Unif(0,t) random variables. Reminder: From a random sample X 1,…X n, the ith order statistic is the ith smallest value, denoted X (i). This makes intuitive sense, because the Poisson process has stationary and independent increments. Thus, we expect the arrival times to be uniformly spread across the interval [0,t]

Applications Electrical engineering-(queueing systems) telephone calls arriving to a system Astronomy-the number of stars in a sector of space, the number of solar flares Chemistry-the number of atoms of a radioactive element that decay Biology-the number of mutations on a given strand of DNA History/war-the number of bombs the Germans dropped on areas of London Famous example (Bortkiewicz)-number of soldiers in the Prussian cavalry killed each year by horse-kicks.

References Grandell, Jan, Mixed Poisson Processes, New York: Chapman and Hall, Hogg, Robert V. and Craig, Allen T., Introduction to Mathematical Statistics, 5 th Ed., Upper Saddle River, New Jersey: Prentice-Hall Inc., 1995, pp Ross, Sheldon M., Introduction to Probability Models, 8 th Ed., New York: Academic Press, pp