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.
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