# Hyperbolic Distribution

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

In this Demonstration we visualize the probability density function of the hyperbolic distribution, which has parameters (location), (tail), (asymmetry), and (scale). These are all real-valued, with the additional constraint that . This distribution has "semi-heavy" tails and has appeared in a diverse range of applications, including models of asset returns in financial markets and sand pile formation. The word "hyperbolic" is used because the log of its probability density function is a hyperbola; this can be seen by clicking the "log scale" checkbox.

Contributed by: Peter Falloon (May 2009)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The probability density function for the hyperbolic distribution is given by

,

where and is the modified Bessel function of the second kind. It has mean and variance .

As , the probability density decays exponentially like . This is intermediate between the behavior of the normal distribution, which decays more rapidly (like ), and the more extreme "fat tail" behavior of power-law distributions. For this reason, it is sometimes referred to as a "semi-heavy tailed" distribution.

Snapshot 1: on a vertical log scale the distribution has a hyperbolic shape

Snapshot 2: as , the hyperbola degenerates into a piecewise linear form

Snapshot 3: as , the hyperbola becomes a parabola

Snapshot 4: for non-zero , the distribution is asymmetric

For more information, see the Wikipedia page for Hyperbolic distribution.

## Permanent Citation

"Hyperbolic Distribution"

http://demonstrations.wolfram.com/HyperbolicDistribution/

Wolfram Demonstrations Project

Published: May 21 2009