# Lec4 - What might account for this discrepancy Binomial...

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What might account for this discrepancy?

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Binomial distribution : what happens when p is small and n is high? P X = k ( ) = k n p k 1 p ( ) n k = n ( n 1)( n 2). ..( n k + 1) k ! p k 1 p ( ) n k = ( pn ) k k ! 1 1 n 1 2 n ... 1 k 1 n 1 p ( ) n k n → ∞ 1 x n 1 P X = k ( ) ( pn ) k k ! 1 p ( ) n k
Binomial distribution : what happens when p is small and n is high? P X = k ( ) ( pn ) k k ! 1 p ( ) n k ( pn ) k k ! 1 p ( ) n 1 p ( ) k P X = k ( ) λ k 1 n n 1 p ( ) k Let us defne = n . p 1 lim n →∞ 1 n n = e As , P X = k ( ) k ! k e

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Binomial distribution : what happens when p is small and n is high? P X = k ( ) λ k ! k e −λ with = n . p If p < 0.1 and n > 50 = Poisson distribution Siméon Denis Poisson (1781-1840)
Poisson Distribution Poisson distribution is a discrete frequency distribution of the number of times a rare event occurs. Two general models lead to a Poisson distribution: 1. Events occurring in space (spatial). Large quantity of some medium in which there is a large total number of discrete small entities . The material is thoroughly mixed so that the small entities are distributed uniformly through the medium. plankton -> seawater bacteria -> petri plate parasites -> host When a small quantity of the medium is sampled the number of the small entities follows the Poisson distribution.

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Poisson Distribution Poisson distribution is a discrete frequency distribution of the number of times a rare event occurs. Two general models lead to a Poisson distribution:
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## This note was uploaded on 11/04/2009 for the course BIO 50935 taught by Professor Bryant during the Fall '09 term at University of Texas.

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Lec4 - What might account for this discrepancy Binomial...

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