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# HyperbolicDistribution

 HyperbolicDistribution represents a hyperbolic distribution with location parameter , scale parameter , shape parameter , and skewness parameter . HyperbolicDistributionrepresents a generalized hyperbolic distribution with shape parameter .
• 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 .
Probability density function of a hyperbolic distribution:
Cumulative distribution function of a hyperbolic distribution:
Mean and variance of a hyperbolic distribution:
Probability density function of a generalized hyperbolic distribution:
Cumulative distribution function of a generalized hyperbolic distribution:
Mean and variance of a generalized hyperbolic distribution:
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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 Scope   (9)
Generate a set of pseudorandom numbers that are hyperbolic distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness of a hyperbolic distribution:
Of a generalized hyperbolic distribution:
Kurtosis of a hyperbolic distribution:
Of a generalized hyperbolic distribution:
Different moments with closed forms as functions of parameters:
Hazard function of a hyperbolic distribution:
Hazard function of a generalized hyperbolic distribution:
Quantile function of a hyperbolic distribution:
Quantile function of a generalized hyperbolic distribution:
 Applications   (5)
A logarithm of the diameter (in millimeters) of mined diamonds follows HyperbolicDistribution with parameters , , , and :
Generate random variates:
Find the probability that the diamond's diameter exceeds 5 millimeters:
Normal inverse Gaussian (NIG) distribution is a special case of HyperbolicDistribution:
It has a particularly simple moment generating function:
Hence the sum of NIG variates also follows NIG distribution:
Fit the daily logarithmic return of the S&P 500 index since 2005 to NIG distribution:
Compare the density of the estimated distribution to a data histogram:
Variance-gamma distribution () is the limiting case of for :
Find the limiting probability density function:
The variance-gamma distribution also admits parameter mixture representation:
Check that densities are equal:
Compare the histogram of the variance-gamma distribution to its PDF:
Skewed distribution is obtained in the limit of :
Find the limiting probability density function:
The skewed distribution also admits parameter mixture representation:
Check that densities are equal:
Compare the histogram of the variance-gamma distribution to its PDF:
Student distribution corresponds to :
The logarithm of population of towns, cities, and villages in the United States can be modeled by HyperbolicDistribution:
Remove missing and zero values:
Find the mean population:
Fit the population logarithm to HyperbolicDistribution:
Compare the data histogram to the fitted density plot:
Parameter influence on the CDF of a hyperbolic distribution for each :
Generalized hyperbolic distribution:
Hyperbolic distribution is closed under translation and scaling by a positive factor:
The logarithm of the PDF of a hyperbolic distribution is a hyperbola:
The logarithm of the PDF can be written as a general hyperbola equation with determinant condition :
The determinant condition is satisfied:
Relationships to other distributions:
Generalized hyperbolic distribution simplifies to hyperbolic distribution:
Generalized hyperbolic distribution is a transformation of NormalDistribution and InverseGaussianDistribution:
It can also be interpreted as ParameterMixtureDistribution:
CauchyDistribution is a singular limit of , given and :
NormalDistribution is the limiting case of for and :
LaplaceDistribution is the limiting case of when and :
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