The hyperbolic distribution is a
continuous probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
characterized by the logarithm of the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
being a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
. Thus the distribution decreases exponentially, which is more slowly than the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. 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 asset
A financial asset is a non-physical asset whose value is derived from a contractual claim, such as bank deposits, bonds, and participations in companies' share capital. Financial assets are usually more liquid than other tangible assets, such ...
s and
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
wind speeds. The hyperbolic distributions form a subclass of the
generalised hyperbolic distributions.
The origin of the distribution is the observation by
Ralph Bagnold
Brigadier Ralph Alger Bagnold, OBE, FRS, (3 April 1896 – 28 May 1990) was an English 20th-century desert explorer, geologist and soldier.
In 1932, he staged the first recorded East-to-West crossing of the Libyan Desert. His work in the fi ...
, 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,
where he also introduced the
generalised hyperbolic distribution, using the fact the a hyperbolic distribution is a random mixture of normal distributions.
References
{{DEFAULTSORT:Hyperbolic Distribution
Continuous distributions