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Slide 1
The Poisson Process Presented by Darrin Gershman and Dave
Wilkerson
Slide 2
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?
Slide 3
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.
Slide 4
Simon Denis Poisson Poissons 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 cole
Centrale and later the cole Polytechnique in Paris, where he
excelled in mathematics, despite having much less formal education
than his peers.
Slide 5
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 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 involving physical topics,
such as pendulums and sound.
Slide 6
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 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 probabilit des jugements 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 his own.
Slide 7
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]
Slide 8
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
Slide 9
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.
Slide 10
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.
Slide 11
Theorem: the alternate definition implies definition 1. Proof:
Fix, and let by independent increments by stationary increments
Assumptions (iii) and (iv) imply
Slide 12
Conditioning on whether N(h) = 0, N(h) = 1, or N(h) 2 implies
As we get, Which is the same as
Slide 13
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.
Slide 14
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
?
Slide 15
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.
Slide 16
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/
)
Slide 17
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.
Slide 18
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.
Slide 19
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.
Slide 20
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]
Slide 21
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.
Slide 22
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://www.worldhistory.com/wiki/P/Poisson-process.htm
http://www.wordiq.com/definition/Poisson_distribution
http://www.quantnotes.com/fundamentals/backgroundmaths/poission.htm
Grandell, Jan, Mixed Poisson Processes, New York: Chapman and Hall,
1997. Hogg, Robert V. and Craig, Allen T., Introduction to
Mathematical Statistics, 5 th Ed., Upper Saddle River, New Jersey:
Prentice-Hall Inc., 1995, pp. 126-8. Ross, Sheldon M., Introduction
to Probability Models, 8 th Ed., New York: Academic Press, pp.
288-322.