Scale relativity

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

Probability density function

A random variable has a Laplace(μ, b) distribution if its probability density function is:

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

$=\frac{1}{2bx}\{\begin{array}{cc}\exp (-\frac{\mu -\ln x}{b})& \text{if }x<\mu \\ \exp (-\frac{\ln x-\mu }{b})& \text{if }x\ge \mu \end{array}$

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

$F(y)=0.5[1+\mathrm{sgn}(\log (y)-\mu )(1-\exp (-|\log (y)-\mu |/b))].$

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.

References

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