A moment of a univariate probability
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
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(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
function can be similarly defined as
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(13)
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Therefore,
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(14)
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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[m, n, ...] in the Mathematica
application package mathStatica
and CentralToCumulant[m, n, ...], respectively.
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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