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Poisson & normal distributions as approximations for the binomial

  1. Jul 17, 2011 #1
    These three quotes talk about the use of the Poisson and normal distributions as approximations for the binomial when n is large. The first two quotes here say Poisson is best when p small, and the normal otherwise. The third seems to change the story; it says Poisson is best for large p too. Is there a contradiction here, or is Hoel just nuancing it as he goes along?

    - Hoel: Introduction to Mathematical Statistics, 5th ed., p. 64.

    - Hoel: ibid., p. 81.

    - Hoel: ibid., p. 85.
  2. jcsd
  3. Jul 17, 2011 #2
    I believe what is intended here is the Poisson distribution works when either p or (1-p) is small, which is what is meant by p small or large. The normal distribution is favored when p~=0.5.

    So, "nuance" it is.
  4. Jul 17, 2011 #3
    The Poisson distribution is a good approximation to the binomial when p << 1. If p is large (i.e., 1-p << 1), then N-n (N = # trials, n = # successes) will follow a Poisson distribution. This is what he means by saying the Poisson "may be used" -- not that n will follow a Poisson, but that you can use the Poisson to calculate the distribution of n. The normal distribution is a good approximation when both Np and N(1-p) are large. Hoel doesn't say this, but when all three conditions are met, p << 1, Np >> 1, and N(1-p) >> 1, both the Poisson and the normal are good approximations. I.e., the normal works even for small p or large p as long as N is big enough to compensate.
  5. Jul 17, 2011 #4
    Excellent! Thanks for clearing that up for me.
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