Poisson Distribution

Author

Eric W. Weisstein
December 27, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Plot

With[{n=10},BarChart[Table[PDF[PoissonDistribution[n],i],{i,0,2n}],BarGroupSpacing->0,PlotRange->All,Ticks->None]
]

Stats

Stats[PoissonDistribution[ν],x]//StatsForm//TraditionalForm

Maximum

PDF[PoissonDistribution[ν],n]

D[ν^n Exp[-ν]/n!,n]

FullSimplify[D[ν^n Exp[-ν]/n!,n]]

Solve[(EulerGamma-HarmonicNumber[n]+Log[ν])==0,n]

Moments

Analytic

V5

Sum[(((E^(-ν)) (ν^k))/(k!)) k^n,{k,0,Infinity}]

Simplify[Table[Sum[(((E^(-ν)) (ν^k))/(k!)) k^n,{k,0,Infinity}],{n,6}]]//ColumnForm

Simplify[Table[Sum[ν^k StirlingS2[n,k],{k,n}],{n,6}]]//ColumnForm

μ[n_]:=Sum[(((E^(-ν)) (ν^k))/(k!)) k^n,{k,0,Infinity}]

FullSimplify[Table[m[n],{n,4}]]

V8

Sum[(((E^(-ν)) (ν^k))/(k!)) k^n,{k,0,Infinity}]

Saslaw Extension

f[N_]:=Nbar(1-b)/N!(Nbar(1-b)+N b)^(N-1)Exp[-Nbar(1-b)-N b]

f[N]

f[N] /. b->0

f[N] /. b->1

Sum[f[N],{N,0,Infinity}]

Sum[Evaluate[f[N] /. b->0],{N,0,Infinity}]

Sum[Evaluate[N f[N] /. b->0],{N,0,Infinity}]

Sum[Evaluate[N^2 f[N] /. b->0],{N,0,Infinity}]

Sum[f[N] N,{N,0,Infinity}]

FullSimplify[Table[Sum[Evaluate[f[N] /. b->i] ,{N,0,Infinity}],{i,0,1,1/4}]]

FullSimplify[Table[Sum[Evaluate[f[N] /. b->i] ,{N,0,Infinity}],{i,0,1,.2}]]

h[s_]:=b+(1-b)/b Sum[N^(N-1)/N! b^N Exp[-N b]s^N,{N,Infinity}]

h[2]

g[N_]:=FullSimplify[f[N] /. Nbar -> ν b/(1-b)]

g[N]

f[N]