TOPICS
Moment
The
thraw moment
(i.e., moment about zero) of a distribution
is defined by
 | (1) |
where
 | (2) |
, themean, is usually simply denoted
. If the moment is instead taken about a point
,
 | (3) |
Astatistical distribution isnot uniquely specified by its moments, although it is by itscharacteristic function.
The moments are most commonly taken about themean. These so-calledcentral moments are denoted
and are defined by
 |  |  | (4) |
 |  |  | (5) |
with
. The second moment about themean is equal to thevariance
 | (6) |
where
is called thestandard deviation.
The relatedcharacteristic function isdefined by
 |  | ![[(d^nphi)/(dt^n)]_(t=0)](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fMoment%2fInline18.svg&f=jpg&w=240) | (7) |
 |  |  | (8) |
The moments may be simply computed using themoment-generatingfunction,
 | (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,VarianceExplore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604-609, 1992.Referenced on Wolfram|Alpha
MomentCite this as:
Weisstein, Eric W. "Moment." FromMathWorld--AWolfram Resource.https://mathworld.wolfram.com/Moment.html