Editing Central moment

{{Short description|Moment of a random variable minus its mean}}
{{Use American English|date = January 2019}}
{{Refimprove|date=September 2014}}
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 characterized. 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 parameter|location]].

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

==Univariate moments==

The {{mvar|n}}-th [[moment (mathematics)|moment]] about the [[mean]] (or {{mvar|n}}-th '''central moment''') of a real-valued [[random variable]] {{mvar|X}} is the quantity {{math|1=''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]] {{math|''f''(''x'')}}, the {{mvar|n}}-th moment about the mean {{mvar|μ}} is<ref name="ProbRand">{{cite book | title=Probability and Random Processes | publisher=Oxford University Press |last1=Grimmett | first1 = Geoffrey | last2 = Stirzaker | first2 = David | year=2009 | location=Oxford, England | isbn=978-0-19-857222-0}}</ref>
<math display="block"> \mu_n = \operatorname{E} \left[ {\left( X - \operatorname{E}[X] \right)}^n \right] = \int_{-\infty}^{+\infty} (x - \mu)^n f(x)\,\mathrm{d} x. </math>

For random variables that have no mean, such as the [[Cauchy distribution]], central moments are not defined.

The first few central moments have intuitive interpretations:
* The "zeroth" central moment {{math|''μ''<sub>0</sub>}} is 1.
* The first central moment {{math|''μ''<sub>1</sub>}} is 0 (not to be confused with the first [[raw moment]] or the [[expected value]] {{mvar|μ}}).
* The second central moment {{math|''μ''<sub>2</sub>}} is called the [[variance]], and is usually denoted {{math|''σ''<sup>2</sup>}}, where {{mvar|σ}} represents the [[standard deviation]].
* The third and fourth central moments are used to define the [[standardized moment]]s which are used to define [[skewness]] and [[kurtosis]], respectively.

===Properties===
For all {{mvar|n}}, the {{mvar|n}}-th central moment is [[Homogeneous function|homogeneous]] of degree {{mvar|n}}:

<math display="block">\mu_n(cX) = c^n \mu_n(X).\,</math>

''Only'' for {{mvar|n}} such that n equals 1, 2, or 3 do we have an additivity property for random variables {{mvar|X}} and {{mvar|Y}} that are [[statistical independence|independent]]:

<math display="block">\mu_n(X+Y) = \mu_n(X)+\mu_n(Y)\,</math> provided ''n'' ∈ {{math|{1, 2, 3}<nowiki/>}}.

A related functional that shares the translation-invariance and homogeneity properties with the {{mvar|n}}-th central moment, but continues to have this additivity property even when {{math|''n'' ≥ 4}} is the {{mvar|n}}-th [[cumulant]] {{math|''κ''<sub>''n''</sub>(''X'')}}. For {{math|1=''n'' = 1}}, the {{mvar|n}}-th cumulant is just the [[expected value]]; for {{mvar|n}}&nbsp;= either 2 or 3, the {{mvar|n}}-th cumulant is just the {{mvar|n}}-th central moment; for {{math|''n'' ≥ 4}}, the {{mvar|n}}-th cumulant is an {{mvar|n}}-th-degree monic polynomial in the first {{mvar|n}} moments (about zero), and is also a (simpler) {{mvar|n}}-th-degree polynomial in the first {{mvar|n}} central moments.

===Relation to moments about the origin===
Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the {{mvar|n}}-th-order moment about the origin to the moment about the mean is

<math display="block">
\mu_n = \operatorname{E}\left[\left(X - \operatorname{E}[X]\right)^n\right]
= \sum_{j=0}^n \binom{n}{j} {\left(-1\right)}^{n-j} \mu'_j \mu^{n-j},
</math>

where {{mvar|μ}} is the mean of the distribution, and the moment about the origin is given by

<math display="block">
\mu'_m = \int_{-\infty}^{+\infty} x^m f(x)\,dx = \operatorname{E}[X^m] = \sum_{j=0}^m \binom{m}{j} \mu_j \mu^{m-j}.
</math>

For the cases {{math|1=''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> and <math>\mu'_0=1</math>):

<math display="block">\mu_2 = \mu'_2 - \mu^2\,</math>
which is commonly referred to as <math> \operatorname{Var}(X) = \operatorname{E}[X^2] - \left(\operatorname{E}[X]\right)^2</math>

<math display="block">\begin{align}
\mu_3 &= \mu'_3 - 3 \mu \mu'_2 +2 \mu^3 \\
\mu_4 &= \mu'_4 - 4 \mu \mu'_3 + 6 \mu^2 \mu'_2 - 3 \mu^4.
\end{align}</math>

... and so on,<ref>{{Cite web|url=http://mathworld.wolfram.com/CentralMoment.html|title = Central Moment}}</ref> following [[Pascal's triangle]], i.e.

<math display="block">\mu_5 = \mu'_5 - 5 \mu \mu'_4 + 10 \mu^2 \mu'_3 - 10 \mu^3 \mu'_2 + 4 \mu^5.\,</math>

because {{nowrap|<math> 5\mu^4\mu'_1 - \mu^5 \mu'_0 = 5\mu^4\mu - \mu^5 = 5 \mu^5 - \mu^5 = 4 \mu^5</math>.}}

The following sum is a stochastic variable having a '''''compound distribution'''''

<math display="block">W = \sum_{i=1}^M Y_i, </math>

where the <math>Y_i</math> are mutually independent random variables sharing the same common distribution and <math>M</math> a random integer variable independent of the <math>Y_k</math> with its own distribution. The moments of <math>W</math> are obtained as

<math display="block">\operatorname{E}[W^n]= \sum_{i=0}^n\operatorname{E}\left[\binom{M}{i}\right] \sum_{j=0}^i \binom{i}{j} {\left(-1\right)}^{i-j} \operatorname{E} \left[ \left(\sum_{k=1}^j Y_k\right)^n \right], </math>

where <math display="inline">\operatorname{E} \left[ {\left(\sum_{k=1}^j Y_k\right)}^n\right] </math> is defined as zero for <math>j = 0</math>.

===Symmetric distributions===

In distributions that are [[symmetric distribution|symmetric about their means]] (unaffected by being [[reflection (mathematics)|reflected]] about the mean), all odd central moments equal zero whenever they exist, because in the formula for the {{mvar|n}}-th moment, each term involving a value of {{mvar|X}} less than the mean by a certain amount exactly cancels out the term involving a value of {{mvar|X}} greater than the mean by the same amount.

==Multivariate moments==

For a [[continuous probability distribution|continuous]] [[Joint probability distribution|bivariate]] [[probability distribution]] with [[probability density function]] {{math|''f''(''x'',''y'')}} the {{math|(''j'',''k'')}} moment about the mean {{math|1=''μ'' = (''μ''<sub>''X''</sub>, ''μ''<sub>''Y''</sub>)}} is
<math display="block"> \begin{align}
\mu_{j,k} &= \operatorname{E} \left[ {\left( X - \operatorname{E}[X] \right)}^j {\left( Y - \operatorname{E}[Y] \right)}^k \right] \\[2pt]
&= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} {\left(x - \mu_X\right)}^j {\left(y - \mu_Y\right)}^k f(x,y) \, dx \, dy.
\end{align} </math>

==Central moment of complex random variables==
The {{mvar|n}}-th central moment for a complex random variable {{mvar|X}} is defined as <ref>{{cite book | last1 = Eriksson | first1 = Jan | last2 = Ollila | first2 = Esa | last3 = Koivunen | first3 = Visa | title = 2009 IEEE International Conference on Acoustics, Speech and Signal Processing | chapter = Statistics for complex random variables revisited | doi = 10.1109/ICASSP.2009.4960396 | pages = 3565–3568 | year = 2009 | isbn = 978-1-4244-2353-8 | s2cid = 17433817 }}</ref>
{{Equation box 1
|indent = :
|equation = <math>\alpha_n = \operatorname{E} \left[ {\left( X - \operatorname{E}[X] \right)}^n \right],</math>
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The absolute {{mvar|n}}-th central moment of {{mvar|X}} is defined as
{{Equation box 1
|indent = :
|equation = <math>\beta_n = \operatorname{E} \left[ {\left|\left( X - \operatorname{E}[X] \right)\right|}^n \right].</math>
|cellpadding= 6
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|border colour = #0073CF
|background colour=#F5FFFA}}

The 2nd-order central moment {{math|''β''<sub>2</sub>}} is called the [[Complex random variable#Variance and pseudo-variance|''variance'']] of {{mvar|X}} whereas the 2nd-order central moment {{math|''α''<sub>2</sub>}} is the [[Complex random variable#Variance and pseudo-variance|''pseudo-variance'']] of {{mvar|X}}.

==See also==
{{div col}}
*[[Standardized moment]]
*[[Image moment]]
*{{Slink|Normal distribution|Moments}}
*[[Complex random variable]]
{{div col end}}

==References==
{{Reflist}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Central Moment}}
[[Category:Statistical deviation and dispersion]]
[[Category:Moments (mathematics)]]

[[fr:Moment (mathématiques)#Moment centré]]