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Log-Laplace distribution - Wikipedia

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Probability distribution

Probability density function

Probability density functions for Log-Laplace distributions with varying parameters

and

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Cumulative distribution function

Cumulative distribution functions for Log-Laplace distributions with varying parameters

and

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

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

f ( x | μ , b ) = 1 2 b x exp ⁡ ( − | ln ⁡ x − μ | b ) {\displaystyle 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 ) − μ ) ( 1 − exp ⁡ ( − | ln ⁡ ( y ) − μ | / b ) ) ] . {\displaystyle F(y)=0.5\,[1+\operatorname {sgn}(\ln(y)-\mu )\,(1-\exp(-|\ln(y)-\mu |/b))].}

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]

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