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| {{Probability distribution |
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| name =hyperbolic|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters =<math>\mu</math> [[location parameter|location]] ([[real number|real]])<br /><math>\alpha</math> <!--to do--> (real)<br /><math>\beta</math> asymmetry parameter (real)<br /><math>\delta</math> [[scale parameter]] (real)<br /><math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>|
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| support =<math>x \in (-\infty; +\infty)\!</math>|
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| pdf =<math>\frac{\gamma}{2\alpha\delta K_1(\delta \gamma)} \; e^{-\alpha\sqrt{\delta^2 + (x - \mu)^2}+ \beta (x - \mu)}</math> <br /><br /><math>K_\lambda</math> denotes a modified Bessel function of the second kind|
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| cdf =<!-- to do -->|
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| mean =<math>\mu + \frac{\delta \beta K_{2}(\delta \gamma)}{\gamma K_1(\delta\gamma)}</math>|
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| median =<!-- to do -->|
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| mode =<math>\mu + \frac{\delta\beta}{\gamma}</math>|
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| variance =<math>\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)</math>|
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| skewness =<!-- to do -->|
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| kurtosis =<!-- to do -->|
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| entropy =<!-- to do -->|
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| mgf =<math>\frac{e^{\mu z}\gamma K_1(\delta \sqrt{ (\alpha^2 -(\beta +z)^2)})}{\sqrt{(\alpha^2 -(\beta +z)^2)}K_1 (\delta \gamma)} </math>|
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| char =<!-- to do -->|
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| }}
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| 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 asset]]s and [[Turbulence|turbulent]] wind speeds. The hyperbolic distributions form a subclass of the [[generalised hyperbolic distribution]]s.
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| 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,<ref>{{cite journal|doi=10.1098/rspa.1977.0041|last=Barndorff-Nielsen|first=Ole|year=1977|title=Exponentially decreasing distributions for the logarithm of particle size|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|volume=353|issue=1674|pages=401–409|jstor=79167|publisher=The Royal Society}}</ref> where he also introduced the [[generalised hyperbolic distribution]], using the fact the a hyperbolic distribution is a random mixture of normal distributions.
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| == References ==
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| <references/>
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| {{ProbDistributions|continuous-infinite}}
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| {{DEFAULTSORT:Hyperbolic Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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