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The Poisson
distribution
SADC Course in Statistics
The Poisson Distribution
• Apply the Poisson Distribution when:
– You wish to count the number of times an
event occurs in a given area of opportunity
– The probability that an event occurs in one
area of opportunity is the same for all areas of
opportunity
– The number of events that occur in one area of
opportunity is independent of the number of
events that occur in the other areas of
opportunity
– The probability that two or more events occur
in an area of opportunity approaches zero as
the area of opportunity becomes smaller
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The Poisson distribution
• The Poisson is a discrete probability
distribution named after a French
mathematician Siméon-Denis Poisson,
1781-1840.
• A Poisson random variable is one that
counts the number of events occurring
within fixed space or time interval.
• The occurrence of individual outcomes are
assumed to be independent of each other.
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The Poisson Distribution
The Poisson distribution is defined by:
f ( x) 
x 
 e
x!
Where f(x) is the probability of x occurrences in
an interval
 is the expected value or mean value of
occurrences within an interval
e is the natural logarithm. e = 2.71828
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Properties of the Poisson Distribution
1. The probability of occurrences is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence of an
event in one interval is independent of an
occurrence on nonoccurrence of an event
in any other interval
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Properties of the Poisson distribution
• The mean of the Poisson distribution is the
parameter .
• The standard deviation of the Poisson
distribution is the square root of . This
implies that the variance of a Poisson
random variable = .
• The Poisson distribution tends to be more
symmetric as its mean (or variance)
increases.
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Examples of data on counts
A common form of data occurring in practice
are data in the form of counts, e.g.
• number of road accidents per year at
different locations in a country
• number of children in different families
• number of persons visiting a given website
across different days
• number of cars stolen in the city each month
An appropriate probability distribution for this
type of random variable is the Poisson
distribution.
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Poisson Distribution Function
• While the number of successes in the
binomial distribution has n as the maximum,
there is no maximum in the case of Poisson.
• This distribution has just one unknown
parameter, usually denoted by  (lambda).
• The Poisson probabilities are determined by
the formula:
P( X  k ) 
k 
e
k!
,
for k  0,1,2,3,
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Example: Mercy Hospital
• Poisson Probability Function
Patients arrive at the
MERCY
emergency room of Mercy
Hospital at the average
rate of 6 per hour on
weekend evenings.
What is the
probability of 4 arrivals in
30 minutes on a weekend evening?
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Example: Mercy Hospital
 Poisson
Probability Function
 = 6/hour = 3/half-hour, x = 4
34 (2.71828)3
f (4) 
 .1680
4!
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MERCY
The Poisson Distribution,
Working with the Equation
• Example: During the 12 p.m. – 1 p.m. noon hour, arrivals
at a curbside banking machine have been found to be
Poisson distributed with a mean of 1.3 persons per minute.
If x = number of arrivals per minute, determine P(X < 2):
–1.3(1.3)0 0.2725
(
2
.
71828
)
P( x  0) 

 0.2725
0!
1
–1.3(1.3)1 0.3543
(
2
.
71828
)
P( x 1) 

 0.3543
1!
1
– P(x < 2) = 0.2725 + 0.3543 =
0.6268
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Basic
Example: Number of cars stolen
• Suppose the number of cars stolen per
month follows a Poisson distribution with
parameter  = 3
What is the probability that in a given month
• Exactly 2 cars will be stolen?
• No cars will be stolen?
• 3 or more cars will be stolen?
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Example: Number of cars stolen
For the first two questions, you will need:
λ 2e 
=
P(X = 2) =
2!
λ 0e 
=
P(X = 0) =
0!
The 3rd is computed as
= 1 – P(X=0) – P(X=1) – P(X=2)
=
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Class Exercise
In example above, we assumed X=family size,
has a Poisson distribution with =5.
Thus P(X=x) = 5x e-5/x! , x=0, 1, 2, …etc.
(a)What is the chance that X=15?
Answer: P(X=15) = 515 e-5/15!
= 0.000157
This is very close to zero. So it would be
reasonable to assume that a family size of 15
was highly unlikely!
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Class Exercise – continued…
(b) What is the chance that a randomly
selected household will have family size < 2 ?
To answer this, note that
P(X < 2) = P(X = 0) + P(X = 1)
=
(c) What is the chance that family size will be
3 or more?
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