Double Exponential (Laplace) distribution

Story

The difference in waiting times for the arrival of a Poisson process is Double-Exponentially (a.k.a. Laplace) distributed (with location parameter \(\mu = 0\)).

Example

The difference in waiting times between repressor-operator binding events.

Parameters

The Double Exponential has a location parameter \(\mu\), which may take on any real value, and a positive scale parameter \(\sigma\).

Support

The Double Exponential distribution is supported on the set of real numbers.

Probability density function

\[\begin{align} f(y;\mu, \sigma) = \frac{1}{\sqrt{2 \sigma}}\,\mathrm{e}^{-\lvert y-\mu\rvert/\sigma}. \end{align}\]

Cumulative distribution function

\[\begin{align} F(y;\mu, \sigma) =\frac{1}{2} + \frac{1}{2}\text{sgn}(y-\mu)\left(1 - \mathrm{e}^{-\lvert y-\mu \rvert/\sigma}\right), \end{align}\]

where \(\text{sgn}(x)\) denotes the sign of \(x\).

Moments

Mean: \(\mu\)

Variance: \(2\sigma^2\)

Usage

Package

Syntax

NumPy

rng.laplace(mu, sigma)

SciPy

scipy.stats.laplace(mu, sigma)

Distributions.jl

Laplace(mu, sigma)

Stan

double_exponential(mu, sigma)

Notes

  • The Double Exponential distribution is often referred to as the Laplace distribution, named for Pierre-Simon.

  • When LASSO regression is considered in a Bayesian context, the priors on the regression parameters are Double Exponetial.

PDF and CDF plots