# Statistics - Laplace Distribution

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 $} \end{cases}$

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.