Central Moment

Author

Eric W. Weisstein
August 10, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Requirements

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

Initialization

<<mathStatica`

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

Binomial Transform

(central1=Table[(Subscript[μ, n])->Plus@@Table[Binomial[n,k](-1)^(n-k) (Subscript[(Overscript[μ, ′]), k])(Subscript[(Overscript[μ, ′]), 1])^(n-k),{k,0,n}],{n,7}]/.{(Subscript[(Overscript[μ, ′]), 0])->1})//ColumnForm

mathStatica

(central2=Table[CentralToRaw[n],{n,7}])//ColumnForm

central1-central2

Central Moments (Subscript[μ, n])in terms of Cumulants (Subscript[κ, n])

mathStatica

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

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

mathStatica

(Subscript[μ, 1,0,…,0])

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

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

(Subscript[μ, n,1])

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

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

mathStatica

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