This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# BetaPrimeDistribution

 BetaPrimeDistribution represents a beta prime distribution with shape parameters p and q. BetaPrimeDistributionrepresents a generalized beta prime distribution with scale parameter . BetaPrimeDistributionrepresents a generalized beta distribution of the second kind with shape parameter .
• The probability density for value in a beta prime distribution is proportional to for .
 Basic Examples   (12)
Probability density function of a beta prime distribution:
Cumulative distribution function for a beta prime distribution:
Mean and variance of a beta prime distribution:
Median of a beta prime distribution:
Probability density function of a generalized beta prime distribution:
Cumulative distribution function of a generalized beta prime distribution:
Mean and variance of a generalized beta prime distribution:
Median of a generalized beta prime distribution:
Probability density function of a generalized beta distribution of the second kind:
Cumulative distribution function of a generalized beta distribution of the second kind:
Mean and variance of a generalized beta distribution of the second kind:
Median of a generalized beta distribution of the second kind:
Probability density function of a beta prime distribution:
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Cumulative distribution function for a beta prime distribution:
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Mean and variance of a beta prime distribution:
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Median of a beta prime distribution:
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Probability density function of a generalized beta prime distribution:
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Cumulative distribution function of a generalized beta prime distribution:
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Mean and variance of a generalized beta prime distribution:
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Median of a generalized beta prime distribution:
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Probability density function of a generalized beta distribution of the second kind:
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Cumulative distribution function of a generalized beta distribution of the second kind:
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Mean and variance of a generalized beta distribution of the second kind:
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Median of a generalized beta distribution of the second kind:
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 Scope   (8)
Generate a set of pseudorandom numbers that are beta prime 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:
For a generalized beta distribution of the second kind, skewness does not depend on :
Kurtosis:
For a generalized beta distribution of the second kind, kurtosis does not depend on :
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Different moments for a generalized beta distribution of the second kind:
Closed form for symbolic order:
Hazard function for a beta prime distribution:
Generalized beta prime distribution:
Generalized beta distribution of the second kind:
Quantile function of a beta prime distribution:
Generalized beta prime distribution:
Generalized beta distribution of the second kind:
 Applications   (2)
BetaPrimeDistribution can be used to model losses:
Remove the clear outlier, Andrew, the most destructive hurricane:
Fit a generalized beta distribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the probability that a loss caused by a hurricane is over 3 billion dollars:
Find the mean hurricane loss:
Simulate possible losses for the next 30 strong hurricanes:
Since BeniniDistribution can also be used to model the losses, compare the fit:
The fit with BetaPrimeDistribution is slightly worse:
BetaPrimeDistribution can be used to model state per-capita incomes:
Fit a generalized beta distribution of the second kind into the data:
Compare the histogram of the data to the PDF of the estimated distribution:
Find the average income per capita:
Find states with income close to the average:
Find the median income per capita:
Find states with income close to the median:
Find the log-likelihood value:
Compare with the fit using DagumDistribution:
Compare with the fit using DavisDistribution:
Compare with the fit using LogLogisticDistribution:
Parameter influence on the CDF for each :
Parameter influence on the CDF of a generalized beta distribution of the second kind:
BetaPrimeDistribution is closed under scaling by a positive factor:
Relations to other distributions:
Beta prime distribution is a special case of the type 6 PearsonDistribution:
Beta prime distribution can be obtained as a transformation of BetaDistribution:
Generalized beta distribution of the second kind is the distribution of the quotient of two independent random variables from GammaDistribution:
Generalized beta of the second kind simplifies to beta prime:
Generalized beta prime is a special case of generalized beta of the second kind:
Generalized beta prime simplifies to beta prime distribution:
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