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Variance-Gamma Distribution


	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.


	Contributed by: Peter Falloon
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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

, 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

See the Wikipedia article on Variance-gamma distribution.

Modified Bessel Function of the Second Kind (Wolfram MathWorld)

"Variance-Gamma Distribution" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/VarianceGammaDistribution/

Contributed by: Peter Falloon

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