Hyperbolic distribution

The hyperbolic distribution is a continuous probability distribution characterized by the logarithm of the probability density function being 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.

hyperbolic
Parameters	$\mu$ location (real) $\alpha$ (real) $\beta$ asymmetry parameter (real) $\delta$ scale parameter (real) $\gamma ={\sqrt {\alpha ^{2}-\beta ^{2}}}$
Support	${\displaystyle x\in (-\infty$
PDF	${\frac {\gamma }{2\alpha \delta K_{1}(\delta \gamma )}}\;e^{-\alpha {\sqrt {\delta ^{2}+(x-\mu )^{2}}}+\beta (x-\mu )}$ $K_{\lambda }$ denotes a modified Bessel function of the second kind
Mean	$\mu +{\frac {\delta \beta K_{2}(\delta \gamma )}{\gamma K_{1}(\delta \gamma )}}$
Mode	$\mu +{\frac {\delta \beta }{\gamma }}$
Variance	${\frac {\delta K_{2}(\delta \gamma )}{\gamma K_{1}(\delta \gamma )}}+{\frac {\beta ^{2}\delta ^{2}}{\gamma ^{2}}}\left({\frac {K_{3}(\delta \gamma )}{K_{1}(\delta \gamma )}}-{\frac {K_{2}^{2}(\delta \gamma )}{K_{1}^{2}(\delta \gamma )}}\right)$
MGF	${\frac {e^{\mu z}\gamma K_{1}(\delta {\sqrt {(\alpha ^{2}-(\beta +z)^{2})}})}{{\sqrt {(\alpha ^{2}-(\beta +z)^{2})}}K_{1}(\delta \gamma )}}$

The origin of the distribution is the observation by Ralph Alger Bagnold, published 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,[1] where he also introduced the generalised hyperbolic distribution, using the fact the a hyperbolic distribution is a random mixture of normal distributions.

References

Barndorff-Nielsen, Ole (1977). "Exponentially decreasing distributions for the logarithm of particle size". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. The Royal Society. 353 (1674): 401–409. doi:10.1098/rspa.1977.0041. JSTOR 79167.

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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