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

subject to the following conditions:

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

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

$X≔\mathrm{RandomVariable}\left(\mathrm{Pareto}\left(a\,c\right)\right)\:$

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

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

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

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

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

$\left\{\begin{array}{cc}\mathrm{undefined}& c\le 1\\ \frac{ca}{c-1}& \mathrm{otherwise}\end{array}\right.$

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

$\left\{\begin{array}{cc}\mathrm{undefined}& c\le 2\\ \frac{c{a}^2}{{\left(c-1\right)}^2\left(c-2\right)}& \mathrm{otherwise}\end{array}\right.$

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.