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 A random variable has a log-Laplace(μ, b) distribution if its probability density function is:
f ( x | μ , b ) = 1 2 b x e − | ln x − μ | b {\displaystyle f(x|\mu ,b)={\frac {1}{2bx}}e^{-{\frac {|\ln x-\mu |}{b}}}}
The cumulative distribution function for Y when y > 0, is
F ( y ) = 0.5 [ 1 + sgn ( ln ( y ) − μ ) ( 1 − e − | ln ( y ) − μ | b ) ] {\displaystyle F(y)=0.5\left[1+\operatorname {sgn} (\ln(y)-\mu )\left(1-e^{-{\frac {|\ln(y)-\mu |}{b}}}\right)\right]}
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.
Properties The mean or expected value of a log-Laplace distributed random variable X with a location parameter μ and a scale parameter b is given by
E ( X ) = e μ 1 − b 2 {\displaystyle E(X)={\frac {e^{\mu }}{1-b^{2}}}}
The variance of X is given by
V a r ( X ) = e 2 μ b 4 + 2 e 2 μ b 2 1 − 4 b 6 + 9 b 4 − 6 b 2 {\displaystyle Var(X)={\frac {e^{2\mu }b^{4}+2e^{2\mu }b^{2}}{1-4b^{6}+9b^{4}-6b^{2}}}}
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