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

# ParetoDistribution

 ParetoDistribution represents a Pareto distribution with minimum value parameter k and shape parameter . ParetoDistributionrepresents a Pareto type II distribution with location parameter . ParetoDistributionrepresents a Pareto type IV distribution with shape parameter .
• The probability density for value in a Pareto distribution is proportional to for , and is zero for . »
 ParetoDistribution[k,] Pareto type I distribution ParetoDistribution[k,,] Pareto type II distribution ParetoDistribution[k,1,,] Pareto type III distribution ParetoDistribution[k,,,] Pareto type IV distribution
• The survival function for value in a Pareto distribution corresponds to:
 ParetoDistribution[k,] ParetoDistribution[k,,] ParetoDistribution[k,,,]
 Basic Examples   (12)
Probability density function for Pareto I:
Cumulative distribution function for Pareto I:
Mean and variance of a Pareto I distribution:
Median of a Pareto I distribution:
Probability density function for Pareto II:
Cumulative distribution function for Pareto II:
Mean and variance of a Pareto II distribution:
Median of a Pareto II distribution:
Probability density function for Pareto IV:
Cumulative distribution function for a Pareto IV distribution:
Mean and variance of a Pareto IV distribution:
Median of a Pareto IV distribution:
Probability density function for Pareto I:
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Cumulative distribution function for Pareto I:
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Mean and variance of a Pareto I distribution:
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Median of a Pareto I distribution:
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Probability density function for Pareto II:
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Cumulative distribution function for Pareto II:
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Mean and variance of a Pareto II distribution:
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Median of a Pareto II distribution:
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Probability density function for Pareto IV:
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Cumulative distribution function for a Pareto IV distribution:
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Mean and variance of a Pareto IV distribution:
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Median of a Pareto IV distribution:
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 Scope   (9)
Generate a set of pseudorandom numbers that are Pareto 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 Pareto I and Pareto II distributions depends only on where defined:
Limiting values:
Skewness of a Pareto IV distribution does not depend on the location parameter :
Kurtosis of Pareto I and Pareto II distributions is the same where defined:
Limiting values:
Kurtosis of a Pareto IV distribution does not depend on the location parameter :
Different moments of a Pareto I distribution with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Different moments of a Pareto II distribution:
Closed form for symbolic order:
Different moments of a Pareto IV distribution:
Hazard function for Pareto I:
Hazard function for Pareto II:
Hazard function for Pareto IV:
Quantile function of a Pareto I distribution:
Quantile function of Pareto II:
Quantile function of Pareto IV:
 Applications   (5)
ParetoDistribution as a long-tailed distribution can be used to model city population sizes:
Compare the histogram of population sizes with the PDF of the estimated distribution:
Find the probability that a city has a population of at least 10000:
Find the mean city size:
Simulate the population size of 20 randomly selected cities:
Use ParetoDistribution to model incomes at a large state university:
Adjust part-time salaries to full-time salaries and select nonzero values:
Fit a Pareto distribution into the data:
Compare the histogram of the data to the PDF of the estimated distribution:
Find average income at the large state university:
Find the probability that a salary is at most \$15000:
Find the probability that a salary is at least \$150000:
Find the median salary:
Simulate the incomes for 100 randomly selected employees of such a university:
The lifetime of a device follows ParetoDistribution:
Find the reliability of the device:
Find the average lifetime of this device:
Find the probability that the device will be operational for more than 6 years:
Find the failure rate of the device:
Consider earthquake magnitudes recorded in the U.S. from 1935 to 1989:
The integer parts of the magnitudes recorded on a Richter scale can be fitted with a ParetoDistribution:
Compare the histogram of the magnitudes with the fitted distribution:
Find the probability of an earthquake with magnitude at least 6 on the Richter scale:
Find the average magnitude:
Simulate the next 30 earthquakes:
Use truncated Pareto IV distribution to define Bradford distribution:
Find the limit of density function when the shape parameter tends to 0:
Substitute to simplify constants:
Cumulative density function:
Mean:
Generate random numbers:
Parameter influence on the CDF of Pareto I distribution for each :
Pareto II:
Pareto IV:
Pareto distribution is closed under translation:
The probability density and random variable have a power-law relationship:
The family of Pareto distribution is closed under the minimum:
For different values of shape parameter:
Special case of truncation:
Relationships to other distributions:
Pareto type II distribution is a special case of type 6 PearsonDistribution:
Pareto type I distribution is a special case of BeniniDistribution:
Pareto II distribution simplifies to Pareto I for :
Pareto IV distribution simplifies to Pareto II for :
Pareto distribution is a distribution of an inverse of PowerDistribution:
A Pareto distribution is the limiting case of the BenktanderGibratDistribution:
A Pareto distribution is the limiting case of the BenktanderWeibullDistribution:
ChiSquareDistribution is a transformation of Pareto-distributed variates:
ChiSquareDistribution is a transformation of Pareto-distributed variates:
Pareto distribution is a transformation of ExponentialDistribution:
The general parameter case:
Transformation of a Pareto distribution yields ExponentialDistribution:
Pareto II is related to BetaPrimeDistribution:
Pareto IV is related to BetaPrimeDistribution:
ParetoDistribution can be obtained as a quotient of ExponentialDistribution and ErlangDistribution:
ParetoDistribution can be obtained as a quotient of ExponentialDistribution and GammaDistribution:
ParetoDistribution is not defined when k is not a real number:
ParetoDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
Pareto II distribution for is not Pareto I:
Pareto II distribution simplifies to Pareto I for :