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Power Function distribution

What it describes

The Power Function distribution is bounded with a CDF that grows as a power of relative position in the interval. It is a simple model for monotone endpoint skewness.

Support is [a,b] with alpha > 0. At alpha=1 it is uniform; values below 1 concentrate density near a, and values above 1 near b.

History and names

The family appears in classical compendia as an elementary Beta case. Its value is pedagogical and descriptive: one power changes a uniform shape.

How it connects to other distributions

It is exactly a rescaled Beta with one parameter equal to 1. It is also a Kumaraswamy special case and can be compared with triangular on bounded intervals.

Where it appears

  • bounded proportions and times with monotone concentration
  • skewed simulation examples and transforms of uniform variables

Fitting cautions

“Power” is used for other families; stating support and CDF avoids ambiguity.

A powered Uniform variable that is also Beta

On the unit interval, Power Function is a Beta law whose second shape equals one. Its quantile is obtained by raising a Uniform variable to a reciprocal power. After rescaling between a and b, this construction retains finite support and concentrates mass toward one endpoint according to alpha.

At alpha=1 it becomes Uniform. Larger values push observations toward the upper bound, while smaller values favour the lower one. Its simplicity makes it a useful benchmark and generator of skewed simulations. That simplicity also limits it: only one monotone direction is available, with no interior peak. A full Beta or Kumaraswamy model is better when both endpoints need separate control.

Decision guide

A good candidate when: a bounded variable changes monotonically toward one endpoint and a single parameter adequately describes that concentration.

Compare it with: Beta, of which it is the rescaled Beta(a,1) case. If both endpoints need independent behaviour or an interior mode appears, use the full family.

References

Power Function Distribution: equations and calculator

Distribution defintion

X∼PowerFunction(α,a,b)X\sim\mathrm{PowerFunction}\left(\alpha,a,b\right)

Distribution domain

x∈[a,b]x\in\left[a,b\right]

Parameters domain and parameters constraints

α∈R+,a∈R,b∈R,a<b\alpha\in\mathbb{R}^{+},a\in\mathbb{R},b\in\mathbb{R},a < b

Cumulative distribution function

FX(x)=(x−ab−a)αF_{X}\left(x\right)=\left(\frac{x-a}{b-a}\right)^{\alpha}

Probability density function

fX(x)=α(x−a)α−1(b−a)αf_{X}\left(x\right)=\frac{\alpha(x-a)^{\alpha-1}}{(b-a)^\alpha}

Percent point function/Sample

FX−1(u)=[a+u(b−a)]−αF^{-1}_{X}\left(u\right)=\left[a+u(b-a)\right]^{-\alpha}

Non-central parametric moments

μk′=E[Xk]=∫abxkfX(x)dx\mu'_{k}=E[X^k]=\int_{a}^{b}x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=a+bαα+1\mathrm{Mean}(X)=\mu'_{1}=\frac{a+b\alpha}{\alpha+1}

Parametric variance

Variance(X)=μ2′−μ1′2=2a2+2abα+b2α(α+1)(α+1)(α+2)−Mean(X)2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{2a^2+2ab\alpha+b^2\alpha(\alpha+1)}{(\alpha+1)(\alpha+2)}-\mathrm{Mean}(X)^{2}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(1−α)α+2α(α+3)\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=2\left(1-\alpha\right)\sqrt{\frac{\alpha+2}{\alpha\left(\alpha+3\right)}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=6(α3−α2−6α+2)α(α+3)(α+4)+3\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{6\left(\alpha^{3}-\alpha^{2}-6\alpha+2\right)}{\alpha\left(\alpha+3\right)\left(\alpha+4\right)}+3

Parametric median

Median(X)=[a+12(b−a)]−α\mathrm{Median}(X)=\left[a+\frac{1}{2}(b-a)\right]^{-\alpha}

Parametric mode

Mode(X)=undefined\mathrm{Mode}(X)=\text{undefined}

Additional information and definitions

a:Location parametera:\text{Location parameter}
b−a:Scale parameterb-a:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}