Zipf Distribution

Author

Eric W. Weisstein
October 31, 2005

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

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

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

Usage

?ZipfDistribution

Stats

Stats[ZipfDistribution[ρ],x]//StatsForm

Normalization

Sum[((x^(-ρ-1))/(Zeta[ρ + 1])),{x,∞}]

CDF

Sum[((x^(-ρ-1))/(Zeta[ρ + 1])),{x,1,x}]//FullSimplify

Raw Moments

Sum[x^n((x^(-ρ-1))/(Zeta[ρ + 1])),{x,∞}]

Table[((Zeta[1-n+ρ])/(Zeta[1+ρ])),{n,0,3}]//FullSimplify

Mean

Sum[x((x^(-ρ-1))/(Zeta[ρ + 1])),{x,∞}]

Central Moments

Sum[(x-((Zeta[ρ])/(Zeta[1+ρ])))^n((x^(-ρ-1))/(Zeta[ρ + 1])),{x,∞}]

Table[Sum[(x-((Zeta[ρ])/(Zeta[1+ρ])))^n((x^(-ρ-1))/(Zeta[ρ + 1])),{x,∞}]/.Zeta->zeta,{n,3}]/.zeta->Zeta//FullSimplify

Apart[%]//TraditionalForm