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CHAPTER 6
POISSON DISTRIBUTION
Outline
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The context
The properties
Notation
Formula
Use of table
Use of Excel
Mean and variance
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POISSON DISTRIBUTION
THE CONTEXT
• An important property of the Poisson distribution:
– The probability of two or more successes will occur in an
interval approaches zero as the interval becomes
smaller.
• Examples where interval is a length of time
– Number of people coming to a bank per hour
– Number of calls received by a telephone operator
per hour
– Number of accidents per week
• Examples where interval is a length of an item
– Number of defects per hundred meter
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POISSON DISTRIBUTION
THE PROPERTIES
• The Poisson distribution has the following properties:
1. The number of successes of various intervals are
independent
2. The probability that a success will occur in an interval is
the same for all intervals of equal size and is
proportional to the size of the interval
3. The probability of two or more successes will occur in
an interval approaches zero as the interval becomes
smaller.
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POISSON DISTRIBUTION
THE NOTATION
• Notation
– x : the number of observed successes
–  : the average number of successes occurring in the
given time interval or region
• A constant
1 1 1 1
e        2.71828
0! 1! 2! 3!
The Excel function for ex is EXP(x)
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POISSON DISTRIBUTION
THE PROBABILITY DISTRIBUTION
• The Poisson probability distribution gives the probability of
getting exactly x successes during a given time interval or
in a specified region
• The probability of getting exactly x successes during a
given time interval or in a specified region is as follows:
P X  x   p x 
e  x

x!
• Note:  is the average number of successes occurring in
the given time interval or region
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POISSON DISTRIBUTION
THE PROBABILITY DISTRIBUTION
Example 1: The marketing manager of a company has noted
that she usually receives 15 complaint calls from
customers during a week (consisting of 5 working days)
and that the calls occur at random. Find the probability of
her receiving exactly 5 calls in a single day.
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POISSON DISTRIBUTION
THE PROBABILITY DISTRIBUTION
Example 2: The marketing manager of a company has noted
that she usually receives 15 complaint calls from
customers during a week (consisting of 5 working days)
and that the calls occur at random. Find the probability of
her receiving at most 2 calls in a single day.
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POISSON DISTRIBUTION
USE OF TABLE
• Table 2, Appendix B, pp. 835-836 gives the probability of
getting at most k successes P X  k  given that  is the
average number of successes occurring in the given time
interval or region
• The table can be used to find the probability of
– exactly k successes: P X  k   P X  k   P X  k 1
– at least k successes: P X  k   1  P X  k 1
– successes between a and b:
Pa  X  b  P X  b  P X  a 1
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POISSON DISTRIBUTION
USE OF TABLE
Example 3: Find the following using Table 1:
P X  2 |   3
P X  4 |   3
Example 4: Find the following using Table 1:
P X  2 |   3
P X  3 |   3
P2  X  4 |   3
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POISSON DISTRIBUTION
USE OF EXCEL
• The Excel function BINOMDIST gives P X  k  and P X  k 
• It takes three arguments. The first 2 arguments are k and 
• The last one is TRUE for P X  k  and FALSE for P X  k 
Example 5 (self study): Find P X  2 |   3 and P X  2 |   3
Verify if Excel gives the same answer as it was obtained
before.
Answer:
P X  2 |   3 = POISSON(2,3,TRUE)
P X  2 |   3 = POISSON(2,3,FALSE)
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POISSON DISTRIBUTION
MEAN AND VARIANCE
• If X is a Poisson random variable, the mean and the
variance of X are:
E X   V  X   
• E(X) is the mean or expected value of X
• V(X) is the variance of X
•  is the average number of successes per period
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READING AND EXERCISES
• Reading: pp. 236-240
• Exercises: 6.82 (use formula and table), 6.84, 6.90
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