Definitions

central-moment

Central moment

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 = leftlangle (X - langle X rangle )^k rightrangle = 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.

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^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 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.

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,text{ and } 4—which are of most interest because of the relations to variance, skewness, and kurtosis, respectively—this formula becomes:

mu_2 = mu'_2 - m^2

mu_3 = mu'_3 - 3 m mu'_2 + 2 m^3

mu_4 = mu'_4 - 4 m mu'_3 + 6 m^2 mu'_2 - 3 m^4.

See also

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