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 pzhon Member Join Date: Mar 2004 Posts: 66 Re: Poisson and Approximating Binomial

A Poisson distribution has one parameter, its mean, m. That is also its variance, so the standard deviation is sqrt(m). When m is large, a Poisson distribution is well-approximated by a normal distribution with the same mean and standard deviation.

When m is small, say 1, or .01, a Poisson distribution is only very poorly approximated by a normal distribution. When you have a binomial distribution with a low mean, it is much more accurate to use a Poisson approximation than a normal approximation.

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3) Is it possible given the information I gave above to provide some range of error for any estimated distribution? Since P(rare event) was estimated from running 30MM trials and seeing 300 occurrences, can we somehow infer some cloud of error around any distribution we come up with?

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That's a statistics question rather than a probability question. You can use the Poisson distributions to find confidence intervals about your observed P, but what is appropriate depends on how you will use P.