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.