HyperbolicDistribution

HyperbolicDistribution[α,β,δ,μ]
represents a hyperbolic distribution with location parameter μ, scale parameter δ, shape parameter α, and skewness parameter β.

HyperbolicDistribution[λ,α,β,δ,μ]
represents a generalized hyperbolic distribution with shape parameter λ.

DetailsDetails

  • The probability density for value in a hyperbolic distribution is proportional to .
  • The probability density for value in a generalized hyperbolic distribution is proportional to  ⅇ^(beta (x-mu))sqrt(delta^2+(x-mu)^2)^(lambda-1/2) TemplateBox[{{lambda, -, {1, /, 2}}, {alpha,  , {sqrt(, {{delta, ^, 2}, +, {{(, {x, -, mu}, )}, ^, 2}}, )}}}, BesselK].
  • HyperbolicDistribution allows α and δ to be any positive real number, λ and μ any real number, and β such that .
  • HyperbolicDistribution can be used with such functions as Mean, CDF, and RandomVariate.

ExamplesExamplesopen allclose all

Basic Examples  (6)Basic Examples  (6)

Probability density function of a hyperbolic distribution:

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Cumulative distribution function of a hyperbolic distribution:

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Mean and variance of a hyperbolic distribution:

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Probability density function of a generalized hyperbolic distribution:

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Cumulative distribution function of a generalized hyperbolic distribution:

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Mean and variance of a generalized hyperbolic distribution:

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Introduced in 2010
(8.0)