A raw moment is a moment 
of a probability function 
taken about 0,
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(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
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(3)
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with 
and 
(Papoulis 1984, p. 146). The first few values are
therefore
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(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
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(8)
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where 
is the characteristic
function, to obtain
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(9)
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The first few terms are then given by
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(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
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(15)
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Therefore,
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(16)
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The multivariate raw moments can be expressed in terms of the multivariate cumulants. For example,
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(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.
