# PowerDistribution

PowerDistribution[k,a]

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

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

## Scope(8)

Generate a sample of pseudorandom numbers from a power distribution:

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:

Closed form for symbolic order:

Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median volume:

## Applications(1)

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

Generate random variates:

Compare sample histogram to the distribution density:

## Properties & Relations(9)

Power distribution is closed under scaling by a positive factor:

Power distribution is closed under Max:

Relationships to other distributions:

KumaraswamyDistribution simplifies to a special case of power distribution:

Power distribution is a transformation of ExponentialDistribution:

ExponentialDistribution can be obtained from power distribution:

Power distribution is a distribution of an inverse of ParetoDistribution:

UniformDistribution is a transformation of PowerDistribution:

PowerDistribution is a special case of PearsonDistribution:

## Neat Examples(1)

PDFs for different a values with CDF contours:

Wolfram Research (2010), PowerDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PowerDistribution.html (updated 2016).

#### Text

Wolfram Research (2010), PowerDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PowerDistribution.html (updated 2016).

#### CMS

Wolfram Language. 2010. "PowerDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/PowerDistribution.html.

#### APA

Wolfram Language. (2010). PowerDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PowerDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_powerdistribution, author="Wolfram Research", title="{PowerDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/PowerDistribution.html}", note=[Accessed: 13-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_powerdistribution, organization={Wolfram Research}, title={PowerDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/PowerDistribution.html}, note=[Accessed: 13-September-2024 ]}