Log-Laplace distribution
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In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = eX has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.
Characterization [edit]
Probability density function [edit]
A random variable has a Laplace(μ, b) distribution if its probability density function is:[1]
![= \frac{1}{2bx}
\left\{\begin{matrix}
\exp \left( -\frac{\mu-\ln x}{b} \right) & \mbox{if }x < \mu
\\[8pt]
\exp \left( -\frac{\ln x-\mu}{b} \right) & \mbox{if }x \geq \mu
\end{matrix}\right.](http://upload.wikimedia.org/math/9/7/f/97f34645438c44cc767081228a6d046c.png)
The cumulative distribution function for Y when y > 0, is
![F(y) = 0.5\,[1 + \sgn(\log(y)-\mu)\,(1-\exp(-|\log(y)-\mu|/b))].](http://upload.wikimedia.org/math/8/8/b/88b0570cffd38fcd50f6e49af02e573a.png)
Versions of the log-Laplace distribution based on an asymmetric Laplace distribution also exist.[2] Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite mean and a finite variance.[2]
References [edit]
- ^ Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time. Cambridge University Press. p. 33. ISBN 978-0-521-83741-5.
- ^ a b Kozubowski, T.J. & Podgorski, K. "A Log-Laplace Growth Rate Model". University of Nevada-Reno. p. 4. Retrieved 2011-10-21.
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