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In lots of applications such as to model the number

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In lots of applications – such as to model the number of patents a firm receives – there are many firms getting zero or one, but some firms get many. In such scenarios, the Poisson distribution is not a very good description. 26
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Geometric Distribution We already saw the geometric distribution for a discrete random variables taking values in 1,2,3,. .. . This is appropriate when the underlying variable is defined to be the trial on which a certain event first happens. The geometric distribution is also used to model the number of events happening in a particular time interval, in which case zero is a possible value. If X 0,1,2,. then it is said to have the Geometric p distribution, with 0 p 1, if f x p 1 p x , x ... 27
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It is easy to show the density sums to one because x 0 1 p x 1/ p . Still using the formula for summing a geometric series (to either directly compute the moments or to obtain the moment generating function) it can be shown that E X 1 p p Var X 1 p p 2 28
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It is often useful to index the density by its mean 0, in which case f x 1 1 1 1 x , x 0,1,2,. .. [because p 1 1 ]. Then Var X 1 2 , and so, relative to the Poisson distribution, the geometric distribution exhibits overdispersion. With 1, P X 6 .0156, which is still small but many times larger than in the Poisson 1 case. The variance for the Geometric 1 is two. 29
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0 .1 .2 .3 .4 .5 f(x) 0 1 2 3 4 5 6 7 8 x PDF for a Geometric(1) Distribution 30
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Negative Binomial Distribution The Poisson and geometric distributions are one-parameter families. This limits their applicability to modeling certain count variables. The negative binomial distribution is more flexible than the Poisson or geometric – it contains the geometric as a special case and effectively contains the Poisson as a special case. Its density can be written many different ways. It is convenient to write the density in terms of its mean 0 and one additional parameter, 0, which we will interpret momentarily. 31
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To write the density in compact form, we need the define the gamma function , defined for any r 0, as Γ r 0 exp u u r 1 du Note that Γ 1 0 exp u du 1. It can be shown that the gamma function satisfies a recurrence relation: Γ r 1 r Γ r , r 0 and so, if k is a positive integer, Γ k k 1 !. 32
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The negative binomial density is a two-parameter family and can be written as f x Γ 1 x Γ 1 x !
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