The Poisson Process

1
The Poisson Process

• Presented by Darrin Gershman and
• Dave Wilkerson

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?

3
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.

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Siméon 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 École Centrale and later the
  École Polytechnique in Paris, where he excelled
  in mathematics, despite having much less formal
  education than his peers.

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Poissons 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.

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Poissons 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 Poissons results to develop his
  own.

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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 gt s, then N(t) ? N(s)
• (d) If t gt s, then N(t)-N(s)the number of
  events in the interval (s,t

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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, st) 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

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Poisson Processes

• Definition 1
• Counting process N(t), t ? 0 is a Poisson
  process with rate ?, ? gt 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, PN(ts) N(s) n
• From condition (iii), we know that N(t) also has
  stationary increments and EN(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.

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Alternate definition of Poisson process

• N(t), t ? 0 is a Poisson process with rate ?, ?
  gt 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 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.

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Theorem the alternate definition implies
definition 1.

• Proof
• Fix , and let
• by independent
  increments
• 
  by stationary increments
• Assumptions (iii) and (iv) imply

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

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• 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 .

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Interarrival times

• We will now look at the distribution of the times
  between events in a Poisson process.
• T1 time of first event in the Poisson process
• T2 time between 1st and 2nd events
• Tn time between (n-1)st and nth events.
• Tn , n1,2, is the sequence of interarrival
  times
• What is the distribution of Tn?

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Distribution of Tn

• First consider T1
• PT1gt t PN(t)0 e-?t (condition (iii) with
  s0, n0)
• Thus, T1 exponential(?)
• Now consider T2
• PT2gtt T1s P0 events in (s,st T1s
• P0 events in (s,st (by stationary
  increments)
• P0 events in (0,t (by independent
  increments)
• PN(t)0 e-?t
• Thus, T2 exponential(?) (same as T1)
• Conclusion The interarrival times Tn, n1,2,
  are iid exponential(?) (mean 1/ ?)
• Thus, we can say that the interarrival times are
  memory less.

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Waiting Times

• We say Sn, n1,2, is the waiting time (or
  arrival time) until the nth event occurs.
• Sn , n ? 1
• Sn is the sum of n iid exponential(?) random
  variables.
• Thus, Sn Gamma(n, 1/?)

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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
• pP(event of type I occurs), 1-pP(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) N1(t), t?0 and N2(t), t?0 are Poisson
  processes with rates ?p and ?(1-p) respectively.
• (3) N1(t), t?0 and N2(t), t?0 are
  independent.
• example males/females
• Poisson processes that have more than 2 types of
  events yield results analogous to those above.

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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) PN(th) N(t) 1 ?(t)h o(h)
• (iv) PN(th) 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.

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Compound Poisson Processes

• Let N(t), t ? 0 be a Poisson process and let
• Yi, 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.

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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 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

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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.

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References

• http//www-gap.dcs.stand.ac.uk/history/Mathematic
  ians/Poisson.html
• http//www-gap.dcs.st-and.ac.uk/history/PictDispl
  ay/Poisson.html
• http//www.worldhistory.com/wiki/P/Poisson-process
  .htm
• http//www.wordiq.com/definition/Poisson_distribut
  ion
• http//www.quantnotes.com/fundamentals/backgroundm
  aths/poission.htm
• Grandell, Jan, Mixed Poisson 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.