Log-Laplace distribution

Log-Laplace distribution
	Probability density function Probability density functions for Log-Laplace distributions with varying parameters and .
	Cumulative distribution function Cumulative distribution functions for Log-Laplace distributions with varying parameters and .

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^X has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.

Characterization

A random variable has a log-Laplace(μ, b) distribution if its probability density function is:^[1]

f(x|\mu ,b)={\frac {1}{2bx}}\exp \left(-{\frac {|\ln x-\mu |}{b}}\right)

The cumulative distribution function for Y when y > 0, is

F(y)=0.5\,[1+\operatorname {sgn}(\ln(y)-\mu )\,(1-\exp(-|\ln(y)-\mu |/b))].

Generalization

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

^ 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" (PDF). University of Nevada-Reno. p. 4. Archived from the original (PDF) on 2012-04-15. Retrieved 2011-10-21.

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[1] Lindsey, J.K. (2004). Statistical analysis of stochastic processes in time. Cambridge University Press. p. 33. ISBN 978-0-521-83741-5.

[growth-2] Kozubowski, T.J. & Podgorski, K. "A Log-Laplace Growth Rate Model" (PDF). University of Nevada-Reno. p. 4. Archived from the original (PDF) on 2012-04-15. Retrieved 2011-10-21.

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[2]

Probability density function Probability density functions for Log-Laplace distributions with varying parameters $\mu$ and $b$ .
Cumulative distribution function Cumulative distribution functions for Log-Laplace distributions with varying parameters $\mu$ and $b$ .