Log Laplace distribution - Alchetron, the free social encyclopedia

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.

Probability density function

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

f ( x | μ , b ) = 1 2 b x exp ⁡ ( − | ln ⁡ x − μ | b ) = 1 2 b x { exp ⁡ ( − μ − ln ⁡ x b ) if x < μ exp ⁡ ( − ln ⁡ x − μ b ) if x ≥ μ

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 ) ) ] .

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.

Differential equation

{ { b x f ′ ( x ) + ( b − 1 ) f ( x ) = 0 , f ( 1 ) = e − μ b 2 b } if x < μ { b x f ′ ( x ) + ( b + 1 ) f ( x ) = 0 , f ( 1 ) = e μ b 2 b } if x ≥ μ

References

Log-Laplace distribution Wikipedia

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