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