Central Moment
المؤلف:
Kendall, M. G.
المصدر:
"The Derivation of Multivariate Sampling Formulae from Univariate Formulae by Symbolic Operation." Ann. Eugenics 10
الجزء والصفحة:
...
18-2-2021
1551
Central Moment
A moment 
of a univariate probability density function 
taken about the mean 
,
where 
denotes the expectation value. The central moments 
can be expressed as terms of the raw moments 
(i.e., those taken about zero) using the binomial transform
 |
(3)
|
with 
(Papoulis 1984, p. 146). The first few central moments expressed in terms of the raw moments are therefore
These transformations can be obtained using CentralToRaw[n] in the Mathematica application package mathStatica.
The central moments 
can also be expressed in terms of the cumulants 
, with the first few cases given by
These transformations can be obtained using CentralToCumulant[n] in the Mathematica application package mathStatica.
The central moment of a multivariate probability density function 
can be similarly defined as
 |
(13)
|
Therefore,
 |
(14)
|
For example,
Similarly, the multivariate central moments can be expressed in terms of the multivariate cumulants. For example,
These transformations can be obtained using CentralToRaw[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/CentralMoment/Inline65.gif" style="height:15px; width:5px" />m, n, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/CentralMoment/Inline66.gif" style="height:15px; width:5px" />] in the Mathematica application package mathStatica and CentralToCumulant[![<span style=]()
{" src="https://mathworld.wolfram.com/images/equations/CentralMoment/Inline67.gif" style="height:15px; width:5px" />m, n, ...![<span style=]()
}" src="https://mathworld.wolfram.com/images/equations/CentralMoment/Inline68.gif" style="height:15px; width:5px" />], respectively.
REFERENCES:
Kendall, M. G. "The Derivation of Multivariate Sampling Formulae from Univariate Formulae by Symbolic Operation." Ann. Eugenics 10, 392-402, 1940.
Kenney, J. F. and Keeping, E. S. "Moments About the Mean." §7.3 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 92-93, 1962.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 146, 1984.
Smith, P. J. "A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa." Amer. Stat. 49, 217-218, 1995
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