Example suppose x normal 29 then p x 1 p x 2 3 1 2 3

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EXAMPLE : Suppose X ~ Normal 2,9 . Then P X 1 P X 2 3 1 2 3 P Z 1/3 P Z 1/3 .629 59
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2 . 4 . Other Distributions Over the Entire Real Line While the normal distribution is by far the most common for modeling random variables that can take on positive and negative values and whose distribution is symmetric about its mean, other distributions are useful. As we saw for the normal, for random variables that range over all of thes idea of location and scale are important. If Z is any random variable with f Z z 0, all z , E Z 0, and Var Z 1, then X Z ,for and 0 has support − , with E X and Var X 2 . 60
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Generally, if we know f Z z and defined X a bZ for b 0, X has pdf f X x 1 b f Z x a b This is a useful device for obtaining random variables with any mean and variance if we are given a basic or standard form of a distribution. (The standard form need not have mean zero or, especially, variance one, but in some cases it does.) 61
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Logistic Distribution We have already seen the CDF and PDF for the standard logistic distribution. If Z has the standard logistic distribution then its density is f Z z exp z / 1 exp z  2 . This density is easily seen to be symmetric about zero by multiplying numerator and denominator by exp z and rewriting the denominator: f Z z 1 exp z  1 exp z  2 1 exp z /2 exp z /2  2 . Also, E | Z | and so E Z 0. With some fancy integration it can be shown that Var Z E Z 2 2 /3 where 3.14159. ... 62
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If we define X Z for and 0 then we get a general logistic random variable. Its PDF is just f X x 1 f Z x exp  x / 1 exp  x /  2 and, clearly, E X Var X 2 2 /3 63
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Sometimes the density is reparameterized so the variance 2 2 2 /3 appears directly, which is obtained by plugging in  /3 . But this more cumbersome expression is not as common. We will not use the logistic distribution much in this class, but it plays an important role in certain econometric models. For the logistic distribution, the excess kurtosis can be shown to be 6/5, which is one way to see that the logistic has “fatter tails” than the normal. Remember, the kurtosis measures are based on standardized random variables.
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EXAMPLE Suppose X Normal 29 Then P X 1 P X 2 3 1 2 3 P Z 13...

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