BUILT-IN MATHEMATICA SYMBOL

PowerDistribution

represents a power distribution with domain parameter k and shape parameter a.

MORE INFORMATION

The probability density for value in a power distribution is proportional to for and zero otherwise.

PowerDistribution allows k and a to be any positive real numbers.

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

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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Scope (7)

Generate a set of pseudorandom numbers that are power distributed:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the shape parameter:

Limiting values:

Kurtosis depends only on the shape parameter:

Limiting values:

Kurtosis attains its minimum:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Applications (1)

Suppose the variance of a normal variate follows PowerDistribution defined on the unit interval. Find the resulting distribution:

Generate random variates:

Compare a sample histogram to the distribution density:

Properties & Relations (7)

Parameter influence on the CDF for each

:

Power distribution is closed under scaling by a positive factor:

Relationships to other distributions:

KumaraswamyDistribution simplifies to a special case of power distribution:

Power distribution is a transformation of ExponentialDistribution:

Power distribution is a distribution of an inverse of ParetoDistribution:

PowerDistribution is a special case of PearsonDistribution: