Hyperbolic distribution

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hyperbolic
Probability density function
Cumulative distribution function
Parameters μ location (real)
α (real)
β asymmetry parameter (real)
δ scale parameter (real)
\gamma = \sqrt{\alpha^2 - \beta^2}
Support x \in (-\infty; +\infty)\!
Probability density function (pdf) \frac{\gamma}{2\alpha\delta K_1(\delta \gamma)} \; e^{-\alpha\sqrt{\delta^2 + (x - \mu)^2}+ \beta (x - \mu)}

Kλ denotes a modified Bessel function of the third kind
Cumulative distribution function (cdf)
Mean \mu + \frac{\delta \beta K_{2}(\delta \gamma)}{\gamma K_1(\delta\gamma)}
Median
Mode \mu + \frac{\delta\beta}{\gamma}
Variance \frac{\delta K_{2}(\delta \gamma)}{\gamma K_1(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left(\frac{K_{3}(\delta\gamma)}{K_{1}(\delta\gamma)} -\frac{K_{2}^2(\delta\gamma)}{K_{1}^2(\delta\gamma)} \right)
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf) \frac{e^{\mu z}\gamma K_1(\delta (\alpha^2 -(\beta +z)^2))}{(\alpha^2 -(\beta +z)^2)K_1 (\delta \gamma)}
Characteristic function

The hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The hyperbolic distributions form a subclass of the generalised hyperbolic distributions.

The origin of the distribution is the observation by Ralph Alger Bagnold in his book The Physics of Blown Sand and Desert Dunes (1941) that the logarithm of the histogram of the empirical size distribution of sand deposits tends to form a hyperbola. This observation was formalised mathematically by Ole Barndorff-Nielsen in a paper in 1977, where he also introduced the generalised hyperbolic distribution, using the fact the a hyperbolic distribution is a random mixture of normal distributions.