The Poisson Distribution

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The Poisson Distribution
We can use the Poisson distribution to estimate
the probability of arrivals at a car wash in one
hour or the number of leaks in 100 miles of
pipeline. Bell Labs uses it to model the arrival
of phone calls.
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The Poisson Distribution
The Poisson distribution is defined by

• 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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Example Mercy Hospital

• Poisson Probability Function
• Patients arrive at the
• 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?

MERCY
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Example Mercy Hospital

• Poisson Probability Function

? 6/hour 3/half-hour, x 4
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Using Excel to ComputePoisson Probabilities

• Formula Worksheet

and so on and so on
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Using Excel to ComputePoisson Probabilities

• Value Worksheet

and so on and so on
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Example Mercy Hospital

• Poisson Distribution of Arrivals

actually, the sequence continues 11, 12,
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Problem 31, p. 229
Consider a Poisson probability distribution with
an average number of occurrences of two per
period.

• Write the appropriate Poisson distribution
• What is the average number of occurrences in
  three time periods?
• Write the appropriate Poisson function to
  determine the probability of x occurrences in
  three time periods.
• Compute the probability of two occurrences in one
  time period.
• Compute the probability of six occurrences in
  three time periods.
• Compute the probability of five occurrences in
  two time periods.

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Problem 31, p. 229
(a)
(b)
(c)
(d)
11
Problem 31, p. 229
(d)
(e)
12
Problem 31, p. 229
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The Hypergeometric Distribution
This is similar to the binominal distribution
except (1) the trials are NOT independent and
(2) the probability of success (?) changes from
trial to trial.
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Hypergeometric Distribution
Let r denote in the population size N labeled a
success. N r is the number of elements in the
population labeled failure.
The hypergeometric distribution is used to
compute the probability that in a random
selection of n elements, selected without
replacement, we obtain x elements labeled success
and N x elements labeled failure.
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Notice that the x successes must be pulled from
the r number of successes in the population and
the n - x failures must be drawn from a
population of N r failures
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Hypergeometric Distribution
Where n the number of trials. N number of
elements in the population r number of elements
in the population labeled a success
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Hypergeometric Distribution
Number of ways a sample of size n -x failures can
be selected from a population of size N -r
Number of ways a sample of size x successes can
be selected from a population of size r
Number of ways a sample of size n can be selected
from a population of size N
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Example Neveready

• Hypergeometric Probability Distribution
• Bob Neveready has removed two
• dead batteries from a flashlight and
• inadvertently mingled them with
• the two good batteries he intended
• as replacements. The four batteries look
  identical.
• Bob now randomly selects two of the four
  batteries. What is the probability he selects
  the two good batteries?

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Example Neveready

• Hypergeometric Probability Distribution

where x 2 number of good batteries
selected n 2 number of batteries
selected N 4 number of batteries in
total r 2 number of good batteries in
total
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Using Excel to ComputeHypergeometric
Probabilities

• Formula Worksheet

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Using Excel to ComputeHypergeometric
Probabilities

• Value Worksheet