Raw Moment

Author

Eric W. Weisstein
October 19, 2003

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

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

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

Requirements

mathStatica Application package
http://www.wolfram.com/products/applications/mathstatica/

Initialization

<<mathStatica`

Raw Moments (Subscript[(Overscript[μ, ′]), n])in terms of Central Moments (Subscript[μ, n])

Binomial Transform

(raw1=Table[(Subscript[(Overscript[μ, ′]), n])->Plus@@Table[Binomial[n,k](Subscript[μ, k])(Subscript[(Overscript[μ, ′]), 1])^(n-k),{k,0,n}],{n,7}]/.{(Subscript[μ, 0])->1,(Subscript[μ, 1])->0})//ColumnForm

mathStatica

(raw2=Table[RawToCentral[n],{n,7}])//ColumnForm

Sort[First[Solve[Equal@@@Table[CentralToRaw[n],{n,5}],Table[(Subscript[(Overscript[μ, ′]), n]),{n,5}]]]]//ColumnForm

raw1-raw2

Raw Moments (Subscript[(Overscript[μ, ′]), n])in terms of Cumulants (Subscript[κ, n])

From Series

museries=Simplify[Series[Exp[Sum[(I t)^k/k! (Subscript[κ, k]),{k,4}]],{t,0,3}]]

mu[n_] := Simplify[SeriesCoefficient[
     Series[Exp[Sum[((I*t)^k/k!)*Subscript[κ, k], 
        {k, n}]], {t, 0, n}], n]*(n!/I^n)]

mu[n_]:=Simplify[SeriesCoefficient[Series[Exp[Sum[(I t)^k/k!(Subscript[κ, k]),{k,n}]],{t,0,n}],n]n!/I^n]

Table[mu[n],{n,5}]//ColumnForm

mathStatica

Table[RawToCumulant[n],{n,7}]//ColumnForm

Multivariate Raw Moments (Subscript[μ, n])in terms of Central Moments  (Subscript[(Overscript[μ, ′]), n])

mathStatica

(Subscript[μ, 1,0,…,0])

?RawToCentral

Table[RawToCentral[{n,0}],{n,5}]//ColumnForm

Sort[First[Solve[Equal@@@Table[CentralToRaw[{n,0}],{n,5}],Table[(Subscript[(Overscript[μ, ′]), n,0]),{n,5}]]]]//ColumnForm

(Subscript[μ, n,1])

Table[RawToCentral[{n,1}],{n,5}]//ColumnForm

Table[CentralToRaw[{n,0}],{n,5}]//ColumnForm

(raw0=Sort[First[Solve[Equal@@@Table[CentralToRaw[{n,0}],{n,5}],Table[(Subscript[(Overscript[μ, ′]), n,0]),{n,5}]]]])//ColumnForm

Table[CentralToRaw[{n,1}],{n,5}]

Sort[First[Solve[Equal@@@Table[CentralToRaw[{n,1}],{n,5}],Table[(Subscript[(Overscript[μ, ′]), n,1]),{n,5}]]]/.raw0]

Multivariate Raw Moments (Subscript[(Overscript[μ, ′]), n,…])in terms of Cumulants (Subscript[κ, n,…])

mathStatica

Table[RawToCumulant[{n,1}],{n,5}]//ColumnForm