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:
, scale parameter 
, shape parameter 
, and skewness parameter 
.
.
in a hyperbolic distribution is proportional to 
.
in a generalized hyperbolic distribution is proportional to 
.
.
: 
with determinant condition 
:
EXAMPLES
Basic Examples 
Scope 
