Statistics 270 - Lecture 11

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Statistics 270 - Lecture 11
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• 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

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Poisson Distribution

• A random variable, X, has a Poisson distribution
  with parameter l (lgt0) if the pmf is
• For x0, 1, 2,

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Poisson Distribution

• Is this a pmf?

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Poisson Distribution

• E(X)
• V(X)

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Poisson Distribution

• Observations

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Example

• Suppose the number of drivers who travel between
  a particular origin and destination during a
  designated time period has a Poisson distribution
  with parameter l20
• Find the probability that ten drivers are
  observed
• Find the probability that at most 10 drivers are
  observed

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Poisson Approximation to the Binomial

• Suppose that the random variable X has a Bin(N,p)
  distribution and n!1 and p!0 so that npl. The
  Bin(n,p) ! Pois(\lambda).
• 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 lnp
• Can us Poisson distribution to compute binomial
  probabilities

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Poisson Approximation to the Binomial

• Proof

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Poisson Approximation to the Binomial

• Rule of thumb

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Example

• A rare blood disease occurs in a population with
  frequency 0.001
• If n people are tested, what is the probability
  that at least two have the rare disease

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Example

• n100
• n1000