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| In [[probability theory]] and [[statistics]], a '''central moment''' is a [[moment_(mathematics)|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 [[moment (mathematics)|moments]], computed in terms of deviations from the mean instead of from the zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its [[location parameter|location]].
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| Sets of central moments can be defined for both univariate and multivariate distributions.
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| ==Univariate moments==
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| The ''n''<sup>th</sup> '''[[moment (mathematics)|moment]] about the [[mean]]''' (or ''n''<sup>th</sup> '''central moment''') of a real-valued [[random variable]] ''X'' is the quantity μ<sub>''n''</sub> := E[(''X'' − E[''X''])<sup>''n''</sup>], where E is the [[expected value|expectation operator]]. For a [[continuous probability distribution|continuous]] [[univariate]] [[probability distribution]] with [[probability density function]] ''f''(''x''), the ''n''<sup>th</sup> moment about the mean μ is | |
| :<math> \mu_n = \operatorname{E} \left[ ( X - \operatorname{E}[X] )^n \right] = \int_{-\infty}^{+\infty} (x - \mu)^n f(x)\,dx. </math>
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| For random variables that have no mean, such as the [[Cauchy distribution]], central moments are not defined.
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| The first few central moments have intuitive interpretations:
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| * The "zeroth" central moment μ<sub>0</sub> is one.
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| * The first central moment μ<sub>1</sub> is zero (not to be confused with the [[Expected value|first moment itself, the expected value or mean]]).
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| * The second central moment μ<sub>2</sub> is called the [[variance]], and is usually denoted σ<sup>2</sup>, where σ represents the [[standard deviation]].
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| * The third and fourth central moments are used to define the [[standardized moment]]s which are used to define [[skewness]] and [[kurtosis]], respectively.
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| ===Properties===
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| The ''n''th central moment is translation-invariant, i.e. for any random variable ''X'' and any constant ''c'', we have
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| :<math>\mu_n(X+c)=\mu_n(X).\,</math>
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| For all ''n'', the ''n''th central moment is [[homogeneous polynomial|homogeneous]] of degree ''n'':
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| :<math>\mu_n(cX)=c^n\mu_n(X).\,</math>
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| ''Only'' for ''n'' such that 1 ≤ ''n'' ≤ 3 do we have an additivity property for random variables ''X'' and ''Y'' that are [[statistical independence|independent]]:
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| :<math>\mu_n(X+Y)=\mu_n(X)+\mu_n(Y)\text{ provided }1 \leq n\leq 3.\,</math>
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| A related functional that shares the translation-invariance and homogeneity properties with the ''n''th central moment, but continues to have this additivity property even when ''n'' ≥ 4 is the ''n''th [[cumulant]] κ<sub>''n''</sub>(''X''). For ''n'' = 1, the ''n''th cumulant is just the [[expected value]]; for ''n'' = either 2 or 3, the ''n''th cumulant is just the ''n''th central moment; for ''n'' ≥ 4, the ''n''th cumulant is an ''n''th-degree monic polynomial in the first ''n'' moments (about zero), and is also a (simpler) ''n''th-degree polynomial in the first ''n'' central moments.
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| ===Relation to moments about the origin===
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| Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the ''n''<sup>th</sup>-order moment about the origin to the moment about the mean is
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| :<math>
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| \mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j \mu'^{n-j},
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| </math>
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| where μ is the mean of the distribution, and the moment about the origin is given by
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| :<math>
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| \mu'_j = \int_{-\infty}^{+\infty} x^j f(x)\,dx.
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| </math> | |
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| 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>\mu = \mu'_1</math>):
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| :<math>\mu_2 = \mu'_2 - \mu^2\,</math>
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| :<math>\mu_3 = \mu'_3 - 3 \mu \mu'_2 + 2 \mu^3\,</math>
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| :<math>\mu_4 = \mu'_4 - 4 \mu \mu'_3 + 6 \mu^2 \mu'_2 - 3 \mu^4.\,</math>
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| ===Symmetric distributions===
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| In a [[symmetric distribution]] (one that is unaffected by being [[reflection (mathematics)|reflected]] about its mean), all odd moments equal zero, because in the formula for the ''n''<sup>th</sup> moment, each term involving a value of ''X'' less than the mean by a certain amount exactly cancels out the term involving a value of ''X'' greater than the mean by the same amount.
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| ==Multivariate moments==
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| For a [[continuous probability distribution|continuous]] [[Joint probability distribution|bivariate]] [[probability distribution]] with [[probability density function]] ''f''(''x'',''y'') the (''j'',''k'') moment about the mean μ = (μ<sub>''X''</sub>, μ<sub>''Y''</sub>) is
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| :<math> \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>
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| ==See also==
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| *[[Standardized moment]]
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| *[[Image moment]]
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| *[[Normal distribution#Moments]]
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| {{Theory of probability distributions}}
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| {{DEFAULTSORT:Central Moment}}
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| [[Category:Statistical deviation and dispersion]]
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| [[Category:Theory of probability distributions]]
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| [[fr:Moment (mathématiques)#Moment centré]]
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Now, many . not permission to get a little obsessive and conduct an inquisition. Be reasonable. But, you to be able to know if thez people you will certainly be working with, especially the managers, intend your idea.
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