Geometric Distribution

Author

Eric W. Weisstein
September 21, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plot

BarChart[Table[PDF[GeometricDistribution[.5],i],{i,1,10}],BarGroupSpacing->0,PlotRange->All,Ticks->None]

Stats

Stats[GeometricDistribution[p],x]//StatsForm//TraditionalForm

CDF

Sum[(((1-p))^k)p,{k,0,n}]//FullSimplify

Normalization

Sum[(((1-p))^k)p,{k,0,∞}]

Sum[q^k,{k,0,∞}]

Raw Moments

V6

$Version

FullSimplify[Sum[p (((1-p))^n)n^k,{n,0,Infinity}]]

Table[p PolyLog[-k,1-p],{k,4}]//FullSimplify

V7

$Version

Sum[p (((1-p))^n)n^k,{n,0,Infinity}]

Sum[p (((1-p))^n)n^k,{n,0,Infinity},Assumptions->k>0]

p==1/2

FullSimplify[p PolyLog[-k,1-p]/. p->1/2]

Table[(1/2) PolyLog[-k,(1/2)],{k,10}]

CoefficientList[Series[-Log[2-Exp[x]],{x,0,10}],x]Table[n!,{n,0,10}]

CoefficientList[Series[Exp[x]/(2-Exp[x]),{x,0,10}],x]Table[n!,{n,0,10}]

CoefficientList[Series[-Log[2-Exp[x]],{x,0,10}],x]Factorial[Range[0,10]]

CoefficientList[Series[((Exp[x])/(2-Exp[x])),{x,0,10}],x]Factorial[Range[0,10]]

Central Moments

V6

$Version

FullSimplify[Sum[p (((1-p))^n)(((n-((1-p)/p)))^k),{n,0,Infinity}]]

Table[p LerchPhi[1-p,-k,((-1+p)/p)],{k,4}]//FullSimplify

%/.{1-p->q,p-1->-q}

V7

Sum[p (((1-p))^n)(((n-((1-p)/p)))^k),{n,0,Infinity}]

Mean and Friends

Through[{Mean,Variance,Skewness,KurtosisExcess}[GeometricDistribution[p]]]

%/.p->1/2

Characteristic Function

CharacteristicFunction[GeometricDistribution[p],t]

Cumulants

CoefficientList[Series[Log[CharacteristicFunction[GeometricDistribution[p],t]],{t,0,4}],t]Factorial[Range[0,4]]/I^Range[0,4]//FullSimplify

-(1-p)D[%,p]//FullSimplify