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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'' = ''e''<sup>''X''</sup> has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.
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| ==Characterization==
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| ===Probability density function===
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| A [[random variable]] has a Laplace(''μ'', ''b'') distribution if its [[probability density function]] is:<ref>{{cite book|title=Statistical analysis of stochastic processes in time|author=Lindsey, J.K.|page=33|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83741-5}}</ref>
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| :<math>f(x|\mu,b) = \frac{1}{2bx} \exp \left( -\frac{|\ln x-\mu|}{b} \right) \,\!</math>
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| ::<math> = \frac{1}{2bx} | |
| \left\{\begin{matrix}
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| \exp \left( -\frac{\mu-\ln x}{b} \right) & \mbox{if }x < \mu
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| \\[8pt]
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| \exp \left( -\frac{\ln x-\mu}{b} \right) & \mbox{if }x \geq \mu
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| \end{matrix}\right.
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| </math>
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| The [[cumulative distribution function]] for ''Y'' when ''y'' > 0, is
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| : <math>F(y) = 0.5\,[1 + \sgn(\log(y)-\mu)\,(1-\exp(-|\log(y)-\mu|/b))].</math> | |
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| Versions of the log-Laplace distribution based on an [[asymmetry|asymmetric]] Laplace distribution also exist.<ref name=growth/> Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite [[mean]] and a finite [[variance]].<ref name=growth>{{cite web|title=A Log-Laplace Growth Rate Model|url=http://wolfweb.unr.edu/homepage/tkozubow/0_logs.pdf|author=Kozubowski, T.J. & Podgorski, K.|page=4|publisher=University of Nevada-Reno|accessdate=2011-10-21}}</ref>
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| ==References==
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| {{reflist}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions with non-finite variance]]
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| {{probability-stub}}
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| [[Category:Probability distributions]]
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