# L10 - Poisson Distribution Flaws occur at random along...

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Poisson Distribution Flaws occur “at random” along length of oil pipeline. Average λ per unit length. Y = number of ﬂaws in a randomly selected sec- tion of pipeline of length 1. Can we determine the distribution of Y ? Generic situation is “incidents” happening over time, instead of ﬂaws over space, for instance earth- quakes of speciﬁed magnitude, or ﬂoods of particular size. 1

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Subdivide pipe (or time) into n small sections of same length. Let p = prob- ability of a ﬂaw in any small section. Assume ﬂaws occur independently in each section. Then Y is binomial ( n,p ). E [ Y ] = np, so to match set np = λ, that is p = λ n . Now let n → ∞ . P ( Y = y ) = ± n y ² p y (1 - p ) n - y = n ! y !( n - y )! λ y n y 1 - λ n ! - y 1 - λ n ! n e - λ λ y y ! as n → ∞ 2
binomial with n large and p small, that is events happening that have a very small probability of occurrence in time or space. Range R Y = { 0 , 1 ,..., } . E

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L10 - Poisson Distribution Flaws occur at random along...

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