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Lectures prepared by: Elchanan Mossel Yelena Shvets

Understand Poisson approximation to binomial distribution with examples. Learn how to calculate probabilities using Poisson distribution. Explore relationship between mean and success rate in large draws.

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Lectures prepared by: Elchanan Mossel Yelena Shvets

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  1. Lectures prepared by:Elchanan MosselYelena Shvets

  2. Poisson approx: bin(1,1) bin(1,1) m = 1, s = 0

  3. Poisson approx: bin(10,1/10) bin(10,1/10) m = 1, s¼ 1

  4. Poisson approx: bin(100,1/100) bin(100,1/100) m = 1, s¼ 1

  5. Poisson approx: bin(1000,1/1000) bin(1000,1/1000) m =1, s¼ 1 Poisson(1)

  6. Poisson Approximation Example: Summer in Arizona: 100 days of Summer; Probability of rain 1/100. P(#RD=0) = (1-1/100)100¼e-1 P(#RD=1) = (100)(1/100)(1-1/100)99 ¼e-1 P(#RD=2) = (100*99/2)(1/100)2(1-1/100)98 ¼ ½(99/100)(100/99)2e-1¼½e-1 P(#RD=k) ¼1/k! e-1

  7. Poisson Approximation to the Binomial Distribution • If p is small and n is large, the distribution of the number of successes in n independent draws is determined by the mean m = np, • according to the Poisson approximation:

  8. The Poisson (m) Distribution • The Poisson distribution with parameter m or Poisson(m) distribution is the distribution of probabilities Pm(k) over {0,1,2,…} defined by:

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