Hyperbolic Distribution

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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



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.

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