Moment

A moment is a sum or integral of a power of a quantity, often taken with respect to a statistical distribution. The th raw moment (i.e., moment about zero) of a distribution is defined by

(1)

where

(2)

, the mean, is usually simply denoted . If the moment is instead taken about a point ,

(3)

A statistical distribution is not uniquely specified by its moments, although it is by its characteristic function.

The moments are most commonly taken about the mean. These so-called central moments are denoted and are defined by

			(4)
			(5)

with . The second moment about the mean is equal to the variance

(6)

where is called the standard deviation.

The related characteristic function is defined by

			(7)
			(8)

The moments may be simply computed using the moment-generating function,

(9)

In analytic number theory, moments often sum or integrate powers of the complex modulus. For example, for a finite family of values , nonnegative weights , and , the weighted sum is a moment of order . If the weights have positive total, dividing by gives an absolute moment for the corresponding discrete statistical distribution. A continuous example is the moment of order of the Riemann zeta function along the critical line, commonly written (Soundararajan 2009).