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BUILT-IN MATHEMATICA SYMBOL
HyperbolicDistribution
HyperbolicDistribution[
, 
, 
, 
]
represents a hyperbolic distribution with location parameter , scale parameter , shape parameter , and skewness parameter .
HyperbolicDistribution[
, , , , ]
represents a generalized hyperbolic distribution with shape parameter .
Details
- 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]](Files/HyperbolicDistribution.en/4.png "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 and to be any positive real number, and any real number, and such that
. - HyperbolicDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Examplesopen all
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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