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Statistics[Distributions]

 Power
 power distribution

 Calling Sequence Power(b, c) PowerDistribution(b, c)

Parameters

 b - scale parameter c - shape parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{c{t}^{c-1}}{{b}^{c}}& t\le b\\ 0& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Power}\left(b,c\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{c}{}{{u}}^{{c}{-}{1}}}{{{b}}^{{c}}}& {u}{\le }{b}\\ {0}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\left\{\begin{array}{cc}\frac{{c}{}{{0.5}}^{{-}{1.}{+}{c}}}{{{b}}^{{c}}}& {0.5}{\le }{b}\\ {0.}& {\mathrm{otherwise}}\end{array}\right\$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{b}{}{c}}{{1}{+}{c}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{b}}^{{2}}{}{c}}{\left({c}{+}{2}\right){}{\left({1}{+}{c}\right)}^{{2}}}$ (4)

References

 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.