Statistics - Laplace Distribution

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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.

Laplace 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} $

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} + \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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