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Statistics[Distributions]

 Pareto
 Pareto distribution

 Calling Sequence Pareto(a, c) ParetoDistribution(a, c)

Parameters

 a - location parameter c - shape parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t

 subject to the following conditions:

$0

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

Examples

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

 See Also