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poisson - THE POISSON DISTRIBUTION The Poisson distribution...

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Unformatted text preview: THE POISSON DISTRIBUTION The Poisson distribution is a limiting case of the binomial distribution which arises when the number of trials n increases indefinitely whilst the product μ = np , which is the expected value of the number of successes from the trials, remains constant. Consider the binomial probability mass function: (1) b ( x ; n, p ) = n ! ( n − x )! x ! p x (1 − p ) n − x . This can be rewritten as (2) μ x x ! n ! ( n − x )! n x ³ 1 − μ n ´ n ³ 1 − μ n ´ − x . The expression may be disassembled for the purpose of taking limits in the component parts. The limits in question are (3) lim( n → ∞ ) n ! ( n − x )! n x = lim( n → ∞ ) ( n ( n − 1) ··· ( n − x + 1) n x ) = 1 , (4) lim( n → ∞ ) ³ 1 − μ n ´ n = e − μ , (5) lim( n → ∞ ) ³ 1 − μ n ´ − x = 1 . On reassembling the parts, it is found that the the binomial function has a limiting form of (6) lim( n → ∞ ) b ( x ; n, p ) = μ x e − μ x !...
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poisson - THE POISSON DISTRIBUTION The Poisson distribution...

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