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 = 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) x x ! n ! ( n x )! n x 1 n n 1 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 ) 1 n n = e , (5) lim( n ) 1 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 ) = x e x !...
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This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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poisson - THE POISSON DISTRIBUTION The Poisson distribution...

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