1、The Poisson ProcessPresented by Darrin Gershman and Dave WilkersonOverview 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
2、process?Simon 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.”Simon Denis Poisson Poissons father originally wanted him to become a doctor. After a brief apprenticeship with an uncle,
3、 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 cole Centrale and later the cole Polytechnique in Paris, where he excelled in mathematics, despite having muc
4、h less formal education than his peers.Poissons education and work Poisson impressed his teachers Laplace and Lagrange with his abilities. Unfortunately, the cole Polytechnique specialized in geometry, and Poisson could not draw diagrams well. However, his final paper on the theory of equations was
5、so good he was allowed to graduate without taking the final examination. After graduating, Poisson received his first teaching position at the cole Polytechnique in Paris, which rarely happened. Poisson did most of his work on ordinary and partial differential equations. He also worked on problems i
6、nvolving physical topics, such as pendulums and sound.Poissons accomplishments Poisson held a professorship at the cole Polytechnique, was an astronomer at the Bureau des Longitudes, was named chair of the Facult des Sciences, and was an examiner at the cole Militaire. He has many mathematical and s
7、cientific tools named for him, including Poissons integral, Poissons equation in potential theory, Poisson brackets in differential equations, Poissons ratio in elasticity, and Poissons constant in electricity. He first published his Poisson distribution in 1837 in Recherches sur la probabilit des j
8、ugements en matire criminelle et matire 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 Poissons results to develop
9、 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,
10、then N(t) N(s)(d) If t s, then N(t)-N(s)=the number of events in the interval (s,tIndependent 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 occurr
11、ing 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 ProcessesDefinition 1:Counting process N(t), t 0 is a Poisson process with rate , 0, if:(i) N(0)=0(ii) N(t) has
12、independent increments(iii) the number of events in any interval of length t Poi(t)( s,t 0, PN(t+s) N(s) = n = From condition (iii), we know that N(t) also has stationary increments and EN(t)= tConditions (i) and (ii) are usually easy to show, but condition (iii) is more difficult to show. Thus, an
13、alternate set of conditions is useful for showing some N(t) is a Poisson process.Alternate definition of Poisson processN(t), t 0 is a Poisson process with rate , 0, if:(i) N(0)=0(ii) N(t) has stationary and independent increments(iii) PN(h) = 1 = h + o(h)(iv) PN(h) 2 = o(h)where function f is said
14、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 letby independent incrementsby stationary incr
15、ementsAssumptions (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 timesWe will now
16、look at the distribution of the times between events in a Poisson process.T1 = time of first event in the Poisson processT2 = time between 1st and 2nd eventsTn = time between (n-1)st and nth events.Tn , n=1,2, is the sequence of interarrival timesWhat is the distribution of Tn?Distribution of TnFirs
17、t consider T1:PT1 t = PN(t)=0 = e-t (condition (iii) with s=0, n=0)Thus, T1 exponential()Now consider T2:PT2t | T1=s = P0 events in (s,s+t | T1=s = P0 events in (s,s+t (by stationary increments) = P0 events in (0,t (by independent increments)= PN(t)=0 = e-t Thus, T2 exponential() (same as T1)Conclus
18、ion: The interarrival times Tn, n=1,2, are iid exponential() (mean 1/ )Thus, we can say that the interarrival times are “memory less.”Waiting TimesWe say Sn, n=1,2, is the waiting time (or arrival time) until the nth event occurs.Sn = , n 1Sn is the sum of n iid exponential() random variables.Thus,
19、Sn Gamma(n, 1/)Poisson processes with multiple types of events Let N(t), t0 be a Poisson process with rate Now partition events into type I, IIp=P(event of type I occurs), 1-p=P(event of type II occurs) N1(t) and N2(t) are the number of type I and type II events Results: (1) N(t) = N1(t) + N2(t)(2)
20、N1(t), t0 and N2(t), t0 are Poisson processes with rates p and (1-p) respectively.(3) N1(t), t0 and N2(t), t0 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 Poiss
21、on 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) PN(t+h) N(t) = 1 = (t)h + o(h)(iv) PN(t+h) N(t) 2 = o(h) Nonhomogeneous Poisson processes are useful when the rate of e
22、vents 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 letYi, i 1 be a family of iid random variables independent of the Poisson process. If
23、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 befor
24、e time t.Order Statistics If N(t) = n, then n events occurred in 0,t Let S1,Sn be the arrival times of those n events. Then the distribution of arrival times S1,Sn is the same as the distribution of the order statistics of n iid Unif(0,t) random variables. Reminder: From a random sample X1,Xn, the i
25、th 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,tApplications Electrical engineering-(queueing systems) telep
26、hone 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 Lo
27、ndon Famous example (Bortkiewicz)-number of soldiers in the Prussian cavalry killed each year by horse-kicks.References http:/www-gap.dcs.stand.ac.uk/history/Mathematicians/Poisson.html http:/www-gap.dcs.st-and.ac.uk/history/PictDisplay/Poisson.html http:/ http:/ http:/ Grandell, Jan, Mixed Poisson
28、Processes, New York: Chapman and Hall, 1997. Hogg, Robert V. and Craig, Allen T., Introduction to Mathematical Statistics, 5th Ed., Upper Saddle River, New Jersey: Prentice-Hall Inc., 1995, pp. 126-8. Ross, Sheldon M., Introduction to Probability Models, 8th Ed., New York: Academic Press, pp. 288-322.
