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64 laplace distribution the density for the standard

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Laplace Distribution The density for the standard Laplace (or double exponential ) distribution is f Z z 1 2 exp | z | , z This is easily seen to be symmetric about zero with a sharp peak at zero. (The function is nondifferentiable at z 0 but infinitely differentiable for all other z .) Later we will compare the tails of this distribution to that of the normal distribution. 65
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0 .1 .2 .3 .4 .5 f(z) -4 -2 0 2 4 z PDF of the Standard Laplace Distribution . range z -4 4 1000 obs was 0, now 1000 . gen fz (1/2)*exp(-abs(z)) . twoway (line fz z) 66
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By integrating separately over z 0 and z 0, the MGF can be showntobe t exp t  1 t 2 1 ,| t | 1. Using the MGF it is easily shown that E Z 0 and E Z 2 Var Z 2. We can shift the location and change the scale by defining X Z , 0 in which case f X x 1 2 exp | x |/ 68
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The Laplace distribution has even thicker tails than the logistic. It can be shown that E Z 4 6 where Z X / and X has the Laplace distribution. So the excess kurtosis (relative to a standard normal) is 3. 69
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Cauchy Distribution The PDF for the standard Cauchy distribution is f Z z 1 1 z 2 , z This PDF is symmetric about zero, so Med Z 0. But the distribution is so spread out that E | Z | , and so E Z is not defined. Var Z is not defined, either. 70
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0 .1 .2 .3 f(z) -5 0 5 z PDF of the Standard Cauchy Distribution 71
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The CDF of the standard Cauchy is F Z z 1 arctan z 1/2. We can generate an entire class of Cauchy distributions as X Z , where Med X is the central tendency. For standard normal, P | Z | 3 .0027. For the standard Cauchy distribution, P | Z | 3 .205, almost 100 times larger than for the standard normal. . di 2*(1 - normal(3)) .0026998 . di 2*(1/2 - (1/_pi)*atan(3)) .20483276 72
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0 .1 .2 .3 .4 -5 0 5 z f(z) Cauchy f(z) normal PDFs for the Standard Cauchy and Standard Normal . range z -5 5 5000 obs was 0, now 5000 . gen fzcauchy 1/(_pi*(1 z^2)) . gen fznormal normalden(z) . twoway (line fzcauchy fznormal z) 73
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2 . 5 . The Lognormal Distribution The lognormal distribution is a continuous distribution over strictly positive values. It is usually defined by starting with a normal random variable, say X ~ Normal , 2 . Then Y exp X has a Lognormal , 2 distribution. Of course, if we start with Y ~ Lognormal , 2 then log Y ~ Normal , 2 . 75
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The density of the lognormal is easily obtained because x g 1 y log y is strictly increasing. So f Y y f X log y  / y 1 y 2 exp log y 2 2 2 1 y  log y / 76
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The expected value of Y is E Y exp 2 /2 exp 2 /2 exp exp .
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64 Laplace Distribution The density for the standard...

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