Central moment

 ( Statistical Deviation And Dispersion ) 

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 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.

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

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Univariate moments The -th moment about the mean (or -th central moment) of a real-valued random variable is the quantity , where E is the expected value. For a continuous univariate probability distribution with probability density function , the -th moment about the mean is

(2025). 9780198572220, Oxford University Press. ISBN 9780198572220

$$\backslash mu\_n\; =\; \backslash operatorname\{E\}\; \backslash left\; \backslash right)\}^n\; \backslash right]\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; (x\; -\; \backslash mu)^n\; f(x)\backslash ,\backslash mathrm\{d\}\; x.$$

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 is 1.
• The first central moment is 0 (not to be confused with the first raw moment or the expected value ).
• The second central moment is called the variance, and is usually denoted , where represents the standard deviation.
• The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

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Properties For all , the -th central moment is homogeneous of degree :

$$\backslash mu\_n(cX)\; =\; c^n\; \backslash mu\_n(X).\backslash ,$$

Only for such that n equals 1, 2, or 3 do we have an additivity property for random variables and that are independent:

$$\backslash mu\_n(X+Y)\; =\; \backslash mu\_n(X)+\backslash mu\_n(Y)\backslash ,$$ provided n ∈ .

A related functional that shares the translation-invariance and homogeneity properties with the -th central moment, but continues to have this additivity property even when is the -th cumulant . For , the -th cumulant is just the expected value; for  = either 2 or 3, the -th cumulant is just the -th central moment; for , the -th cumulant is an -th-degree monic polynomial in the first moments (about zero), and is also a (simpler) -th-degree polynomial in the first central moments.

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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 -th-order moment about the origin to the moment about the mean is

$$\backslash mu\_n\; =\; \backslash operatorname\{E\}\backslash left\backslash left(X\backslash right)^n\backslash right]\; =\; \backslash sum\_\{j=0\}^n\; \backslash binom\{n\}\{j\}\; \{\backslash left(-1\backslash right)\}^\{n-j\}\; \backslash mu\text{'}\_j\; \backslash mu^\{n-j\},$$

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

$$\backslash mu\text{'}\_m\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; x^m\; f(x)\backslash ,dx\; =\; \backslash operatorname\{E\}X^m\; =\; \backslash sum\_\{j=0\}^m\; \backslash binom\{m\}\{j\}\; \backslash mu\_j\; \backslash mu^\{m-j\}.$$

For the cases — which are of most interest because of the relations to variance, skewness, and kurtosis, respectively — this formula becomes (noting that $\backslash mu\; =\; \backslash mu\text{'}\_1$ and $\backslash mu\text{'}\_0=1$):

$$\backslash mu\_2\; =\; \backslash mu\text{'}\_2\; -\; \backslash mu^2\backslash ,$$ which is commonly referred to as $\backslash operatorname\{Var\}(X)\; =\; \backslash operatorname\{E\}X^2\; -\; \backslash left(\backslash operatorname\{E\}X\backslash right)^2$

... and so on, following Pascal's triangle, i.e.

$$\backslash mu\_5\; =\; \backslash mu\text{'}\_5\; -\; 5\; \backslash mu\; \backslash mu\text{'}\_4\; +\; 10\; \backslash mu^2\; \backslash mu\text{'}\_3\; -\; 10\; \backslash mu^3\; \backslash mu\text{'}\_2\; +\; 4\; \backslash mu^5.\backslash ,$$

because

The following sum is a stochastic variable having a compound distribution

$$W\; =\; \backslash sum\_\{i=1\}^M\; Y\_i,$$

where the $Y\_i$ are mutually independent random variables sharing the same common distribution and $M$ a random integer variable independent of the $Y\_k$ with its own distribution. The moments of $W$ are obtained as

$$\backslash operatorname\{E\}W^n=\; \backslash sum\_\{i=0\}^n\backslash operatorname\{E\}\backslash left\backslash binom\{M\}\{i\}\backslash right\; \backslash sum\_\{j=0\}^i\; \backslash binom\{i\}\{j\}\; \{\backslash left(-1\backslash right)\}^\{i-j\}\; \backslash operatorname\{E\}\; \backslash left,$$

where $\backslash operatorname\{E\}\; \backslash left$ is defined as zero for $j\; =\; 0$.

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

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Multivariate moments For a continuous bivariate probability distribution with probability density function the moment about the mean is

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Central moment of complex random variables The -th central moment for a complex random variable is defined as

(2025). 9781424423538 ISBN 9781424423538

The absolute -th central moment of is defined as

The 2nd-order central moment is called the variance of whereas the 2nd-order central moment is the pseudo-variance of .

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See also

• Standardized moment
• Image moment
• Complex random variable

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