In this Demonstration we visualize the probability density function of the variance-gamma distribution, which has parameters (location), (tail), (asymmetry), and (scale). These are all real-valued, with the additional constraints , . This distribution has "semi-heavy" tails and has appeared in a diverse range of applications, including models of asset returns in financial markets and turbulent wind speeds.

The probability density function for the variance-gamma distribution is given by

,

where 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: as increases, the distribution becomes more rounded around its peak value

Snapshot 2: for non-zero , the distribution becomes skewed, that is, asymmetric

Snapshot 3: as increases, the tails drop off more steeply