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Binomial and Poisson Derivations watch

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    Can anybody please explain the derivations of the binomial and poisson distributions ?
    thanks alot
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    Define derivations
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    How to you reach the equations for the distributions (how was it created ... )
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    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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