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In probability theory and statistics, the k^{th} moment about the mean (or k^{th} central moment) of a real-valued random variable X is the quantity μ_{k} := E[(X − E[X])^{k}], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x) the moment about the mean μ is
## Properties

The nth central moment is translation-invariant, i.e. for any random variable X and any constant c, we have## 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 n^{th}-order moment about the origin to the moment about the mean is## See also

- $$

Note that $langle\; X\; rangle$ is equivalent to E(X) (i.e the expectation of X); it is the notation preferred by physicists.

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 first central moment is zero.
- The second moment about the mean is called the variance, and is usually denoted σ
^{2}, where σ represents the standard deviation. - The third and fourth moments about the mean are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

- $mu\_n(X+c)=mu\_n(X).,$

For all n, the nth central moment is homogeneous of degree n:

- $mu\_n(cX)=c^nmu\_n(X).,$

Only for n ≤ 3 do we have an additivity property for random variables X and Y that are independent:

- $mu\_n(X+Y)=mu\_n(X)+mu\_n(Y)\; mathrm\{provided\}\; nleq\; 3.,$

A related functional that shares the translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κ_{n}(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ≥ 4, the nth cumulant is an nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-degree polynomial in the first n central moments.

- $$

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

- $$

For the cases $n=2,3,text\{\; and\; \}\; 4$—which are of most interest because of the relations to variance, skewness, and kurtosis, respectively—this formula becomes:

- $mu\_2\; =\; mu\text{'}\_2\; -\; m^2$

- $mu\_3\; =\; mu\text{'}\_3\; -\; 3\; m\; mu\text{'}\_2\; +\; 2\; m^3$

- $mu\_4\; =\; mu\text{'}\_4\; -\; 4\; m\; mu\text{'}\_3\; +\; 6\; m^2\; mu\text{'}\_2\; -\; 3\; m^4.$

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Last updated on Friday September 26, 2008 at 08:53:14 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 26, 2008 at 08:53:14 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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