Central Moment

A moment of a univariate probability density function taken about the mean ,

			(1)
			(2)

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

			(4)
			(5)
			(6)
			(7)
			(8)

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

			(9)
			(10)
			(11)
			(12)

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,

			(15)
			(16)

Similarly, the multivariate central moments can be expressed in terms of the multivariate cumulants. For example,

			(17)
			(18)
			(19)
			(20)
			(21)

These transformations can be obtained using CentralToRaw[m, n, ...] in the Mathematica application package mathStatica and CentralToCumulant[m, n, ...], respectively.