Handout 26
Supplements material in section 6.4 of L-G
The Poisson Process
The Poisson process is a probability model for an experiment in which events occur randomly
and uniformly in time or space. There are three assumptions, which we will first describe for
random events in time.
Independence The numbers of events in non overlapping time intervals are independent. Thus
the number of events in time interval
is independent of the number of events in
for any
.
Individuality Events occur singly rather than in pairs or groups. This means that if we choose
small enough, the probability of getting 2 or more events in time interval
is
negligible in comparison with the probability of getting 0 or 1 events.
Homogeneity Events occur at a uniform rate over the entire time period being considered. Thus
the expected number of events in a time interval is proportional to its length. The expected
number of events in any time interval of length is where is a constant. , the expected
number of events per unit time, is called the intensity parameter of the process.
It can be shown that, if these three conditions hold, then the number of events in a time interval of
length has a Poisson distribution with mean
, that is
events in a time interval of length
for
(1)
Example 26.1 Highway Accidents We might use a Poisson process to model the occurrence of
car accidents along a stretch of a highway. The first assumption implies that the number of accidents between 5:10 and 5:20 should be independent of the number of accidents between 5:00 and
5:10. Anyone involved in an accident at 5:05 is unlikely to be at another at 5:15 and so we will
not have strict independence. However, this should not be a serious problem if there are a large
number of motorists.
In order that the second assumption holds, we would have to treat a collision involving 2 or
more vehicles as a single event. The second assumption would not hold if we considered damage
to a vehicle or injury to a passenger as an event because then events would tend to occur in groups
rather that singly.
In order for the homogeneity assumption to hold, we would have to restrict attention to a time
period during which traffic volume and weather conditions were relatively constant. Otherwise the
intensity parameter (accident rate) would not be constant over the entire time period.
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36-217: Probability Theory and Random Processes
Fall 1997
If the three assumptions hold, then the probability that there are accidents during a time
interval of length will be given by (1), where is the average of expected number of accidents
per unit time.
Example 26.2 During the busy period from 4:00 to 6:00pm on weekdays, accidents occur along a
stretch of highway at the average rate of 3 per hour. What is the probability that there will be more
than 6 accidents in a 1-hour period? What is the probability that there will be at most 2 accidents
during the 2-hour busy period on a particular day?
Solution. The assumptions of a Poisson process should hold to a good approximation; see the
discussion in Example 26.1. Thus the probability of accidents in hours will be given by (1)
with
accidents per hour. The probability of accidents in one hour is thus
for
The probability of more than 6 accidents in one hour is then
The expected number of accidents in 2 hours is
hours is
, and the probability of
accidents in 2
for
The probability of at most 2 accidents in 2 hours is then
Trick for calculating Poisson probabilities recursively
Observe that
That is, if you have already calculated
you can easily get
2
by taking
36-217: Probability Theory and Random Processes
Fall 1997
Random events in space
The Poisson process is also used as a model for experiments in which events occur randomly
and uniformly in space. The three assumptions are
Independence The number of events in non overlapping regions are independent.
Individuality Events occur singly rather than in pairs or groups
Homogeneity Events occur at a uniform rate over the entire region being considered.
For random events in the plane, the probability of
events in a region of are
is
for
where is the average or expected number of events per unit area. For random events in 3-D space,
the probability of events in a region of volume is
for
where is the expected number of events per unit volume.
A Poisson process in the plane might be a suitable model for the distribution of points of impact
of meteorites on the earths surface, the distribution of bacterial colonies over a glass plate, or the
distribution of surface flaws on a metal sheet. A Poisson process in three-dimensional space might
be used to describe the distribution of bacteria in river water, of the distribution of flaws in a solid.
Example 26.3 During World War II, the city of London was subjected to flying-bomb attacks.
The technology at that time did not allow bombs to be aimed at particular targets, but only at the
general vicinity. Consequently, the points of impact of flying bombs should have the appearance
of “random points in the plane”. The number of hits in a region of area should have a Poisson
distribution with mean
, where is the expected number of hits per unit area.
0
1
2
3
4
229
211
93
35
7
1
226.74 211.39 98.54 30.62 7.14 1.57
Table 1: Observed and Expected Frequencies of Flying-Bomb Hits
The number of flying-bomb hits was determined for each of 576 regions of equal area in South
London. there were 537 hits altogether, and thus the average number of hits per region was
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36-217: Probability Theory and Random Processes
Fall 1997
. Table 1 gives , the observed number of regions with hits, and , the
expected number under a Poisson distribution with mean 0.9323. the fit of the Poisson distribution
to these data is extremely good.
Extra Credit Problem/ Due Next Time
Customers arrive at a soft drink dispensing machine according to a Poisson process with a
rate . Suppose that each time a customer deposits money, the machine dispenses a soft drink
with probability . Find the probability distribution function for the number of soft drinks (
)
dispensed in time . (Assume that the machine holds an infinite number of soft drinks).
Next Time: Markov Chains
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