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
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
The moments may be simply computed using the moment-generating
function,
 |
(9)
|
See also
Absolute Moment,
Characteristic Function,
Charlier's Check,
Cumulant-Generating
Function,
Factorial Moment,
Kurtosis,
Mean,
Moment-Generating
Function,
Moment Problem,
Moment
Sequence,
Skewness,
Standard
Deviation,
Standardized Moment,
Variance Explore this topic in the
MathWorld classroom
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References
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill,
pp. 145-149, 1984.Press, W. H.; Flannery, B. P.; Teukolsky,
S. A.; and Vetterling, W. T. "Moments of a Distribution: Mean, Variance,
Skewness, and So Forth." §14.1 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 604-609, 1992.Referenced on Wolfram|Alpha
Moment
Cite this as:
Weisstein, Eric W. "Moment." From MathWorld--A
Wolfram Web Resource. https://mathworld.wolfram.com/Moment.html
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