Central Moment

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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.



References

2016

[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 = (μX, μY) 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

[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).