Log-Laplace distribution

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 &mu; and b, then Y = eX has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.

Characterization
A random variable has a log-Laplace(&mu;, b) distribution if its probability density function is:


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

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


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

Generalization
Versions of the log-Laplace distribution based on an asymmetric Laplace distribution also exist. Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite mean and a finite variance.