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Double Exponential (Laplace) Distribution

NIST/SEMATECH Section 1.3.6.6.12 Double Exponential Distribution

PDF

Laplace(0, 1) PDF -6 -4 -2 0 2 4 6 x 0 0.1 0.2 0.3 0.4 0.5 f(x)

CDF

Laplace(0, 1) CDF -6 -4 -2 0 2 4 6 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)=12βexp⁡ ⁣(−∣x−μ∣β)f(x) = \frac{1}{2\beta}\exp\!\left(-\frac{|x - \mu|}{\beta}\right)

Cumulative Distribution Function

F(x)={12exp⁡ ⁣(x−μβ)x<μ1−12exp⁡ ⁣(−x−μβ)x≥μF(x) = \begin{cases} \frac{1}{2}\exp\!\left(\frac{x-\mu}{\beta}\right) & x < \mu \\ 1 - \frac{1}{2}\exp\!\left(-\frac{x-\mu}{\beta}\right) & x \ge \mu \end{cases}

Properties

Mean

μ\mu

Variance

2β22\beta^2

Overview

The double exponential (Laplace) distribution is a symmetric distribution centred at μ\mu with scale β\beta, having heavier tails than the normal distribution. It is the distribution of the difference of two independent exponential random variables.

Related Distributions

Normal Distribution Exponential Distribution Cauchy Distribution