BUILT-IN MATHEMATICA SYMBOL

ParetoDistribution

represents a Pareto distribution with minimum value parameter k and shape parameter

.

ParetoDistribution

represents a Pareto type II distribution with location parameter

.

ParetoDistribution

represents a Pareto type IV distribution with shape parameter

.

MORE INFORMATION

The probability density for value in a Pareto distribution is proportional to for , and is zero for . »

ParetoDistribution includes Pareto distributions of types I, II, III, and IV:

	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

ParetoDistribution is also known as Lomax distribution.

The survival function for value in a Pareto distribution corresponds to:

	ParetoDistribution[k,]
	ParetoDistribution[k,,]
	ParetoDistribution[k,,,]

ParetoDistribution allows k, , and to be any positive real numbers and any real number.

ParetoDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

EXAMPLES

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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:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Different moments of a Pareto II distribution:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

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:

Define Bradford distribution:

Bradford probability density function:

Cumulative density function:

Mean:

Generate random numbers:

Properties & Relations (21)

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:

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:

Possible Issues (3)

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

: