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Laplace distribution represents the distribution of differences between two independent variables having identical exponential distributions. It is also called double exponential distribution.
Probability density function
Probability density function of Laplace distribution is given as:
Formula
${ L(x | mu, b) = frac{1}{2b} e^{- frac{| x – mu |}{b}} }$
$ { = frac{1}{2b} } $
$ begin {cases}
e^{- frac{x – mu}{b}}, & text{if $x lt mu $} \[7pt]
e^{- frac{mu – x}{b}}, & text{if $x ge mu $}
end{cases} $
$ begin {cases}
e^{- frac{x – mu}{b}}, & text{if $x lt mu $} \[7pt]
e^{- frac{mu – x}{b}}, & text{if $x ge mu $}
end{cases} $
Where −
-
${mu}$ = location parameter.
-
${b}$ = scale parameter and is > 0.
-
${x}$ = random variable.
Cumulative distribution function
Cumulative distribution function of Laplace distribution is given as:
Formula
${ D(x) = int_{- infty}^x}$
$ = begin {cases}
frac{1}{2}e^{frac{x – mu}{b}}, & text{if $x lt mu $} \[7pt]
1- frac{1}{2}e^{- frac{x – mu}{b}}, & text{if $x ge mu $}
end{cases} $
frac{1}{2}e^{frac{x – mu}{b}}, & text{if $x lt mu $} \[7pt]
1- frac{1}{2}e^{- frac{x – mu}{b}}, & text{if $x ge mu $}
end{cases} $
$ { = frac{1}{2} + frac{1}{2}sgn(x – mu)(1 – e^{- frac{| x – mu |}{b}}) } $
Where −
-
${mu}$ = location parameter.
-
${b}$ = scale parameter and is > 0.
-
${x}$ = random variable.
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