Log Series Distribution

Author

Eric W. Weisstein
November 2, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/Log-SeriesDistribution.nb.

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

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

Plot

With[{θ=.95},
BarChart[Table[PDF[LogSeriesDistribution[θ],i],{i,1,20}],BarGroupSpacing->0,PlotRange->All,Ticks->None]
]

Stats

FullSimplify[Stats[LogSeriesDistribution[θ],n]]//StatsForm//TraditionalForm

Normalization

Sum[-θ^n/(n Log[1-θ]),{n,∞}]

CDF

v4.2

Sum[-t^x/(x Log[1-t]),{x,n}]

FullSimplify[%]

Through[{LeafCount,ByteCount}[%]]

v5.0

Sum[-t^x/(x Log[1-t]),{x,n}]

FullSimplify[%]

Through[{LeafCount,ByteCount}[%]]

V7

Sum[-t^x/(x Log[1-t]),{x,n}]

FullSimplify[%]

Equivalence

FullSimplify[((Beta[t,1+n,0])/(Log[1-t]))-(((t^(1+n)) Gamma[1+n] Hypergeometric2F1Regularized[1+n,1,2+n,t])/(Log[1-t]))]

FullSimplify[1+((Beta[t,1+n,0])/(Log[1-t]))-1+(((t^(1+n)) LerchPhi[t,1,1+n])/(Log[1-t]))]

Table[1+((Beta[t,1+n,0])/(Log[1-t]))-(1+(((t^(1+n)) LerchPhi[t,1,1+n])/(Log[1-t]))),{n,0,10}]//FullSimplify

Moments

Raw Moments

Sum[-θ^i/(i Log[1-θ])i^n,{i,∞}]

Table[-((PolyLog[1-n,θ])/(Log[1-θ])),{n,4}]//FullSimplify

Central Moments

Sum[-θ^i/(i Log[1-θ])(i-(θ/((-1+θ) Log[1-θ])))^n,{i,∞}]

Mean

Sum[-θ^i/(i Log[1-θ])i,{i,∞}]

Variance