Poisson Distributions

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Poisson Distributions

• This distribution deals with the probabilities of
  rare events that occur infrequently in space,
  time, distance, area, etc.
• Examples
• The number of accidents that occur per month at a
  given intersection
• The number of tardies per semester for a given
  student
• The number of runs per inning in a baseball game

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Properties

• The occurrence of a success in any interval is
  independent of that in any other interval
• The probability that a success will occur in any
  interval is the same for all intervals of equal
  size and is proportional to the size of the
  interval
• We observe a discrete number of events in a
  continuous (fixed) interval.

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Formulas
X number of rare events per unit of time,
space, etc. l mean value of X (Greek letter
lambda)
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The number of accidents in an office building
during a four-week period averages 2. What is
the probability there will be one accident in the
next four-week period?
What is the probability that there will be more
than two accidents in the next four-week period?
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From 800 until 830 is a 30 minute period. From
800 until 900 is a 60 minute period. Since the
period is doubled, you must double the mean
amount of calls to keep it proportional!

• The number of calls to a police department
  between 8 pm and 830 pm on Friday averages 3.5.
• What is the probability of no calls during this
  period?
• What is the probability of no calls between 8 pm
  and 9 pm on Friday night?
• What is the mean and standard deviation of the
  number of calls between 10 pm and midnight on
  Friday night?

P(X 0) poissonpdf(3.5,0) .0302
Be sure to adjust l!
P(X 0) poissonpdf(7,0) .0009
m 14 s 3.742
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Examine the histograms of the Poisson
distributions l 2 l 4
What happens to the shape?
What happens to the means?
What happens to the standard deviations?
l 6
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As l increases

• The distributions become more symmetrical
• The means increase
• The standard deviations increase