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Moment

The nth raw moment mu_n^' (i.e., moment about zero) of a distribution P(x) is defined by

mu_n^'=<x^n>,
(1)

where

<f(x)>={sumf(x)P(x) discrete distribution; intf(x)P(x)dx continuous distribution.
(2)

mu_1^', the mean, is usually simply denoted mu=mu_1. If the moment is instead taken about a point a,

mu_n(a)=<(x-a)^n>=sum(x-a)^nP(x).
(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 mu_n and are defined by

mu_n=<(x-mu)^n>
(4)
=int(x-mu)^nP(x)dx,
(5)

with mu_1=0. The second moment about the mean is equal to the variance

mu_2=sigma^2,
(6)

where sigma=sqrt(mu_2) is called the standard deviation.

The related characteristic function is defined by

phi^((n))(0)=[(d^nphi)/(dt^n)]_(t=0)
(7)
=i^nmu_n(0).
(8)

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

mu_n^'=M^((n))(0).
(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.

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Moment

Cite this as:

Weisstein, Eric W. "Moment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Moment.html

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