Logarithmic Distribution

Author

Eric W. Weisstein
January 14, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Definition

LogDistribution /:Domain[LogDistribution[a_,b_]]:=Interval[{a,b}]
LogDistribution /: PDF[LogDistribution[a_,b_],x_]:=Log[x]/(b(Log[b]-1)-a(Log[a]-1))
LogDistribution /: CDF[LogDistribution[a_,b_],x_]:=((a-x-a Log[a]+x Log[x])/(a-b-a Log[a]+b Log[b]))
LogDistribution /: Mean[LogDistribution[a_,b_]]:=(((a^2)-(b^2)-2 (a^2) Log[a]+2 (b^2) Log[b])/(4 a-4 b-4 a Log[a]+4 b Log[b]))

Stats

Stats[LogDistribution[a,b],x]//StatsForm//TraditionalForm

Plot

With[{a=1,b=5},Show[GraphicsArray[Plot[#[LogDistribution[a,b],x],{x,a,b},AxesLabel->TraditionalForm/@{x,#[x]},
DisplayFunction->Identity,
Ticks->None,PlotRange->All,PlotStyle->Red]&/@{PDF,CDF}],
ImageSize->500
]]

Normalization

Integrate[PDF[LogDistribution[a,b],x],{x,a,b},Assumptions->0<a<b]

CDF

Integrate[PDF[LogDistribution[a,b],x],{x,a,x},Assumptions->0<a<x<b]

Raw Moments

Integrate[(x^n) PDF[LogDistribution[a,b],x],{x,a,b},Assumptions->0<a<b]//Simplify

Together[%/. n->1]

Mean

Integrate[x PDF[LogDistribution[a,b],x],{x,a,b},Assumptions->0<a<b]//Factor

Variance

Integrate[(((x-(((a^2)-(b^2)-2 (a^2) Log[a]+2 (b^2) Log[b])/(4 a-4 b-4 a Log[a]+4 b Log[b]))))^2) PDF[LogDistribution[a,b],x],{x,a,b},Assumptions->0<a<b]//FullSimplify