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Log-Laplace distribution
Probability density function
Probability density functions for Log-Laplace distributions with varying parameters
μ
{\displaystyle \mu }
and
b
{\displaystyle b}
.
Cumulative distribution function
Cumulative distribution functions for Log-Laplace distributions with varying parameters
μ
{\displaystyle \mu }
and
b
{\displaystyle b}
.
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
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))].}
Generalization
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]
References
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint)
Directional
Degenerate and
singular Families