Moment

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

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

where

$$\left\langle{f(x)}\right\rangle{} =\cases{ \sum f(x)P(x) & d... ...stribution\cr \int f(x)P(x)\,dx & continuous distribution.\cr}$$ (2)

$\mu_1'$, the Mean, is usually simply denoted $\mu=\mu_1$. If the moment is instead taken about a point $a$,

$$\mu_n(a) = \left\langle{(x-a)^n}\right\rangle{} = \sum (x-a)^nP(x).$$ (3)

The moments are most commonly taken about the Mean. These moments are denoted $\mu_n$ and are defined by

$$\mu_n \equiv \left\langle{(x-\mu)^n}\right\rangle{},$$ (4)

with $\mu_1=0$. The moments about zero and about the Mean are related by

$$\mu_2 = \mu_2'-(\mu_1')^2$$ (5)

$$\mu_3 = \mu_3'-3\mu_2'\mu_1'+2(\mu_1')^3$$ (6)

$$\mu_4 = \mu_4'-4\mu_3'\mu_1'+6\mu_2'(\mu_1')^2-3(\mu_1')^4.$$ (7)

The second moment about the Mean is equal to the Variance

$$\mu_2=\sigma^2,$$ (8)

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

The related Characteristic Function is defined by

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

The moments may be simply computed using the Moment-Generating Function,

$$\mu_n'=M^{(n)}(0).$$ (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

References

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.