Laplace Distribution

A sharp-peaked, fat-tailed alternative to the Normal Distribution.

The "Double Exponential" Distribution

The Laplace distribution is a continuous probability distribution that is notable for its sharper peak at the mean and its "fatter" tails compared to the Normal distribution. This means it assigns higher probability to values near the mean and also to extreme outlier events.

In finance and machine learning, this makes it a valuable tool. It can model financial returns that are more prone to extreme events than a normal model would suggest. It is also intrinsically linked to LASSO (L1) regularization, a popular technique in regression for feature selection, because its shape naturally encourages some parameters to go to zero.

The Formula

The probability density function (PDF) is given by:

f(x | \mu, b) = \frac{1}{2b} \exp\left( -\frac{|x - \mu|}{b} \right)

$\mu$ (mu) is the location parameter, which is also the mean, median, and mode.
$b > 0$ is the scale parameter, which controls the spread or "width" of the distribution. A larger b results in a wider, flatter curve.

Interactive Laplace Distribution

Adjust the location (μ) and scale (b) parameters to see how the distinctive shape of the distribution changes.

Location (μ): 0.0

Scale (b): 1.0