Central Moment

A Central Moment is a moment-generating function that is based on the statistical deviations of a probability distribution from the mean value or zero.

• See: Moment-Generating Function, Kurtosis Measure, Skewness Measure, Normal Distribution, Binomial Distribution, Statistical Deviation, Statistical Dispersion.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Central_moment Retrieved 2016-08-21
  • In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterised. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

    Sets of central moments can be defined for both univariate and multivariate distributions.

    Univariate moments: The nth moment about the mean (or nth central moment) of a real-valued random variable X is the quantity μ_n := E[(X − E[X])ⁿ], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x), the nth moment about the mean μ is

[math]\displaystyle{ \mu_n = \operatorname{E} \left[ (X - \operatorname{E}[X] )^n \right] = \int_{-\infty}^{+\infty} (x - \mu)^n f(x)\,\mathrm{d} x. }[/math]

(...)Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the nth-order moment about the origin to the moment about the mean is

[math]\displaystyle{ \mu_n = \mathrm{E}\left[\left(X - \mathrm{E}\left[X\right]\right)^n\right] = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j \mu^{n-j}, }[/math]

where μ is the mean of the distribution, and the moment about the origin is given by

[math]\displaystyle{ \mu'_j = \int_{-\infty}^{+\infty} x^j f(x)\,dx = \mathrm{E}\left[X^j\right] }[/math]

For the cases n = 2, 3, 4 — which are of most interest because of the relations to variance, skewness, and kurtosis, respectively — this formula becomes (noting that [math]\displaystyle{ \mu = \mu'_1 }[/math] and [math]\displaystyle{ \mu'_0=1 }[/math])(...)

Multivariate moments: For a continuous bivariate probability distribution with probability density function f(x,y) the (j,k) moment about the mean μ = (μ_X, μ_Y) is

[math]\displaystyle{ \mu_{j,k} = \operatorname{E} \left[ (X - \operatorname{E}[X] )^j (Y - \operatorname{E}[Y] )^k \right] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (x - \mu_X)^j (y - \mu_Y)^k f(x,y )\,dx \,dy. }[/math]

2016

• (Weisstein, 2016) ⇒ Weisstein, Eric W. "Central Moment." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/CentralMoment.html Retrieved 2016-08-21
  • A moment [math]\displaystyle{ \mu_n }[/math] of a univariate probability density function P(x) taken about the mean [math]\displaystyle{ \mu=\mu_1' }[/math],

[math]\displaystyle{ \mu_n=\lt (x-\lt x\gt )^n\gt =\int(x-\mu)^nP(x)dx }[/math]

where <X> denotes the expectation value. The central moments [math]\displaystyle{ \mu_n }[/math] can be expressed as terms of the raw moments [math]\displaystyle{ mu_n' }[/math] (i.e., those taken about zero) using the binomial transform

[math]\displaystyle{ \mu_n=\sum_{k=0}^n(n; k)(-1)^{n-k}\mu_k'\mu_1'^{n-k} }[/math]

with [math]\displaystyle{ \mu_0'=1 }[/math] (Papoulis 1984, p. 146).

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