A moment of a probability function taken about 0,
(1)
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(2)
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The raw moments (sometimes also called "crude moments") can be expressed as terms of the central moments (i.e., those taken about the mean ) using the inverse binomial transform
(3)
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with and (Papoulis 1984, p. 146). The first few values are therefore
(4)
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(5)
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(6)
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(7)
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The raw moments can also be expressed in terms of the cumulants by exponentiating both sides of the series
(8)
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where is the characteristic function, to obtain
(9)
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The first few terms are then given by
(10)
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(11)
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(12)
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(13)
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(14)
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These transformations can be obtained using RawToCumulant[n] in the Mathematica application package mathStatica.
The raw moment of a multivariate probability function can be similarly defined as
(15)
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Therefore,
(16)
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The multivariate raw moments can be expressed in terms of the multivariate cumulants. For example,
(17)
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(18)
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These transformations can be obtained using RawToCumulant[m, n, ...] in the Mathematica application package mathStatica.