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Statistics 270 - Lecture 11
• Last day/Today: More discrete probability distributions
• Assignment 4: Chapter 3: 5, 7,17, 25, 27, 31, 33, 37, 39, 41, 45, 47,
51, 65, 67, 77, 79
Poisson Distribution
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A random variable, X, has a Poisson distribution with parameter l (l>0) if
the pmf is
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For x=0, 1, 2, …
Poisson Distribution
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Is this a pmf?
Poisson Distribution
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E(X)
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V(X)
Poisson Distribution
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Observations:
Example
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Suppose the number of drivers who travel between a particular origin and
destination during a designated time period has a Poisson distribution with
parameter l=20
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Find the probability that ten drivers are observed
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Find the probability that at most 10 drivers are observed
Poisson Approximation to the Binomial
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Suppose that the random variable X has a Bin(N,p) distribution and n!1
and p!0 so that np=l. The Bin(n,p) ! Pois(\lambda).
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So, in any binomial experiment where n is large and p is small then the
binomial distribution is approximately the same as the Poisson distribution
where l=np
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Can us Poisson distribution to compute binomial probabilities
Poisson Approximation to the Binomial
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Proof:
Poisson Approximation to the Binomial
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Rule of thumb:
Example
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A rare blood disease occurs in a population with frequency 0.001
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If n people are tested, what is the probability that at least two have the
rare disease
Example
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n=100
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n=1000