Central moment

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In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity μk := 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

\mu_k = \left\langle ( X - \langle X \rangle )^k \right\rangle = \int_{-\infty}^{+\infty} (x - \mu)^k f(x)\,dx.

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 σ2, 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.

[edit] Properties

The nth central moment is translation-invariant, i.e. for any random variable X and any constant c, we have

\mu_n(X+c)=\mu_n(X).\,

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

\mu_n(cX)=c^n\mu_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}\ n\leq 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 κn(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.

[edit] 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 nth-order moment about the origin to the moment about the mean is

\mu_n = \sum_{j=0}^n {n \choose j} (-1) ^{n-j} \mu'_j m^{n-j},

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

\mu'_j = \int_{-\infty}^{+\infty} x^j f(x)\,dx.


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

μ2 = μ'2 − m2
μ3 = μ'3 − 3mμ'2 + 2m3
μ4 = μ'4 − 4mμ'3 + 6m2μ'2 − 3m4

[edit] See also