Pareto Distribution

Author

Eric W. Weisstein
September 12, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/ParetoDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ParetoDistribution.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Usage

?ParetoDistribution

Plot

Show[GraphicsArray[
Plot[Evaluate[Table[#[ParetoDistribution[1,a],x],{a,1,5,2}]],{x,1,5},DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{x,#[x]},Ticks->None,PlotStyle->{Black,Red,Blue}]&/@{PDF,CDF}
]]

Stats

Stats[ParetoDistribution[b,a],x]//StatsForm//FullSimplify//TraditionalForm

Moments

Raw Moments

Integrate[x^n a (b^a) (x^(-1-a)),{x,b,Infinity},Assumptions->b>0&&a>n]//Timing

Table[Integrate[x^n a (b^a) (x^(-1-a)),{x,b,Infinity},Assumptions->{b>0&&Re[a]>4}],{n,4}]

Central Moments

Integrate[(x-((a b)/(-1+a)))^n a (b^a) (x^(-1-a)),{x,b,Infinity},Assumptions->b>0&&a>n≥1&&n∈Integers]//Timing

FunctionExpand[a (b^n) Gamma[a-n] Hypergeometric2F1Regularized[a-n,-n,1+a-n,(a/(-1+a))]]

FullSimplify[%]

Table[a (b^n) Gamma[a-n] Hypergeometric2F1Regularized[a-n,-n,1+a-n,(a/(-1+a))],{n,4}]//FunctionExpand//Factor

({(Subscript[μ, 1]),(Subscript[μ, 2]),(Subscript[μ, 3]),(Subscript[μ, 4])}=FullSimplify[Table[Integrate[(x-((a b)/(-1+a)))^n a (b^a) (x^(-1-a)),{x,b,Infinity},Assumptions->{b>0&&Re[a]>4}],{n,4}]])//TableForm

Skewness

FullSimplify[PowerExpand[(Subscript[μ, 3])/((Subscript[μ, 2])^(3/2))]]

Kurtosis

Simplify[PowerExpand[(Subscript[μ, 4])/((Subscript[μ, 2])^2)-3]]

Characteristic Function

FullSimplify[PowerExpand[FourierTransform[a (b^a) (x^(-1-a)) UnitStep[x-b],x,t]]]