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1 1 1 1 x x 012 when 1 we get the geometric density

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1 1 1 1 x , x 0,1,2,. ... When 1 we get the geometric density [because Γ 1 1 and Γ 1 x x !]. Can show that, as 0, the density converges to the Poisson density. It can be shown that E X Var X 1   2 33
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Thus, we can write Var X E X 1  and so is a measure of overdispersion. For a given ,as increases, Var X / E X increases. Later, when we turn to statistics, we will see that if we fix (for example, 1 in the geometric) – that is, we assume we know it – then estimating from a sample of data is easy. The problem is considerably harder when and both need to be estimated. We will write X ~ NegBin , . 34
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The NB density can have a mode different from zero quite generally. 0 .05 .1 .15 f(x) 0 1 2 5 10 15 20 x NegBin PDF with E(X) = 5 and Var(X) = 55 E(X) 35
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2 . COMMON CONTINUOUS DISTRIBUTIONS We have covered the uniform distribution on an interval a , b and also the exponential distribution. We provide several additional distributions here, but the list is not exhaustive. 2 . 1 . Distributions for Unbounded , Nonnegative Random Variables We have already used the exponential distribution. The shape of its density is restricted by its dependence on a single parameter. There is a useful generalization. Gamma Distribution As with many distributions, there are many ways to parameterize the gamma distribution. It is useful to have one of the parameters be the 36
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mean, , and the other a measure of dispersion, which we call . 37
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As with the NegBin distribution, the density of the gamma distribution depends on the gamma function: f x 1  1 Γ 1 x 1 1 exp x /   , x 0 When 1 we get the Exponential distribution. This parameterization is such that Var X  2 , which allows us to introduce the notion of overdisperson 1 and underdispersion 1 relative to the exponential distribution. 38
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A more common parameterization is, for , 0, f x 1 Γ x 1 exp x / , x 0 where E X  Var X  2 The density is easier to look at but a parameterization in terms of the mean is very useful for more complicated econometric modeling. 39
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Pareto Distribution For parameters , 0, the general Pareto , density is given by f x 1 x / 1 , x 0 This density can take on a variety of shapes depending on the values of the parameters.
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1 1 1 1 x x 012 When 1 we get the geometric density because...

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