slides_3_distributions

# 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
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 shown to be 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