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The $n$th moment of a distribution about zero $\mu_n'$ is defined by

\mu_n' = \left\langle{x^n}\right\rangle{},
\end{displaymath} (1)

\left\langle{f(x)}\right\rangle{} =\cases{ \sum f(x)P(x) & d...
...stribution\cr \int f(x)P(x)\,dx & continuous distribution.\cr}
\end{displaymath} (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) = \left\langle{(x-a)^n}\right\rangle{} = \sum (x-a)^nP(x).
\end{displaymath} (3)

The moments are most commonly taken about the Mean. These moments are denoted $\mu_n$ and are defined by
\mu_n \equiv \left\langle{(x-\mu)^n}\right\rangle{},
\end{displaymath} (4)

with $\mu_1=0$. The moments about zero and about the Mean are related by
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle \mu_2'-(\mu_1')^2$ (5)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle \mu_3'-3\mu_2'\mu_1'+2(\mu_1')^3$ (6)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle \mu_4'-4\mu_3'\mu_1'+6\mu_2'(\mu_1')^2-3(\mu_1')^4.$ (7)

The second moment about the Mean is equal to the Variance
\end{displaymath} (8)

where $\sigma=\sqrt{\mu_2}$ is called the Standard Deviation.

The related Characteristic Function is defined by

\phi^{(n)}(0) \equiv \left[{d^n\phi\over dt^n}\right]_{t = 0} = i^n\mu_n(0).
\end{displaymath} (9)

The moments may be simply computed using the Moment-Generating Function,
\end{displaymath} (10)

A Distribution is not uniquely specified by its moments, although it is by its Characteristic Function.

See also Characteristic Function, Charlier's Check, Cumulant-Generating Function, Factorial Moment, Kurtosis, Mean, Moment-Generating Function, Skewness, Standard Deviation, Standardized Moment, Variance


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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© 1996-9 Eric W. Weisstein