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# Binomial and Poisson Derivations watch

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1. Can anybody please explain the derivations of the binomial and poisson distributions ?
thanks alot
2. Define derivations
3. How to you reach the equations for the distributions (how was it created ... )
4. Binomial(n, p)
Consider n independent trials, each with probability p of success. There are nCm ways of getting exactly m successes; each possibility has probability p^m (1 - p)^(n - m). So

P(exactly m successes) = nCm p^m (1 - p)^(n - m)

--

Poisson(mu)
Poisson(mu) is defined to be a binomial distribution with mean mu and a very large number of trials.

For a concrete example, Poisson(1) is roughly Binomial(10^10, 10^-10). You do something that is very unlikely to succeed a very large number of times, then count the successes.

Think also about the number of winners in this week's national lottery draw.

For any nonnegative integer m, for all n >= m,

P(Binomial(n, mu/n) = m)
= nCm (mu/n)^m (1 - mu/n)^(n - m)
= [(1/m!)n(n - 1)(n - 2) ... (n - m + 1)] (mu/n)^m (1 - mu/n)^(-m) (1 - mu/n)^n
= (1/m!) mu^m [n(n - 1)(n - 2) ... (n - m + 1)/n^m] (1 - mu/n)^(-m) (1 - mu/n)^n
-> (1/m!) mu^m * 1 * 1 * e^(-mu)
= (1/m!) mu^m e^(-mu) . . . . . as n -> infinity

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Updated: January 15, 2005
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