The power distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The power variate with scale parameter 1 and shape parameter c is related to the Beta variate with first scale parameter c and second scale parameter 1 by Power(1,c) ~ Beta(c,1).

The power variate with scale parameter b and shape parameter 1 is equivalent to the Uniform variate with lower bound parameter 0 and upper bound parameter b: Power(b,1) ~ Uniform(0,b).

Note that the Power command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Power}\left(b\,c\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{c{u}^{c-1}}{b^c}& u\le b\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\left\{\begin{array}{cc}\frac{c{0.5}^{-1.+c}}{b^c}& 0.5\le b\\ 0.& \mathrm{otherwise}\end{array}\right.$

$\mathrm{Mean}\left(X\right)$

$\frac{bc}{1+c}$

$\mathrm{Variance}\left(X\right)$

$\frac{b^{2}c}{\left(c+2\right){\left(1+c\right)}^2}$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.