Central moment

Template:Short description Template:Use American English Template:Refimprove 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.

Univariate moments

The Template:Mvar-th moment about the mean (or Template:Mvar-th central moment) of a real-valued random variable Template:Mvar is the quantity μ_n := E[(X − E[X])ⁿ], where E is the expectation operator. For a continuous univariate probability distribution with probability density function f(x), the Template:Mvar-th moment about the mean Template:Mvar is $${\mu }_n=\mathrm{E}\left[{(X-\mathrm{E}[X])}^n\right]=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}(x-\mu {)}^{n}f(x)\mathrm{d}x.$$

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 μ₀ is 1.
• The first central moment μ₁ is 0 (not to be confused with the first raw moment or the expected value Template:Mvar).
• The second central moment μ₂ is called the variance, and is usually denoted σ², where Template:Mvar 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.

Properties

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

$${\mu }_n(cX)=c^n{\mu }_n(X).$$

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

$${\mu }_n(X+Y)={\mu }_n(X)+{\mu }_n(Y)$$ provided n ∈ {1, 2, 3}.

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

$${\mu }_n=\mathrm{E}\left[{(X-\mathrm{E}[X])}^n\right]=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j}){(-1)}^{n-j}\mu {\text{'}}_j{\mu }^{n-j},$$

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

$$\mu {\text{'}}_m=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{x}^{m}f(x)dx=\mathrm{E}[X^m]=\sum _{j=0}^m(\genfrac{}{}{0ex}{}{m}{j}){\mu }_j{\mu }^{m-j}.$$

For the cases n = 2, 3, 4 — which are of most interest because of the relations to variance, skewness, and kurtosis, respectively — this formula becomes (noting that $\mu =\mu {\text{'}}_1$ and $\mu {\text{'}}_0=1$):

$${\mu }_2=\mu {\text{'}}_2-{\mu }^2$$ which is commonly referred to as $\mathrm{Var}(X)=\mathrm{E}[X^2]-{(\mathrm{E}[X])}^2$

$$\begin{array}{rl}{\mu }_3& =\mu {\text{'}}_3-3\mu \mu {\text{'}}_2+2{\mu }^3\\ {\mu }_4& =\mu {\text{'}}_4-4\mu \mu {\text{'}}_3+6{\mu }^2\mu {\text{'}}_2-3{\mu }^4.\end{array}$$

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

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

because $5{\mu }^4\mu {\text{'}}_1-{\mu }^5\mu {\text{'}}_0=5{\mu }^4\mu -{\mu }^5=5{\mu }^5-{\mu }^5=4{\mu }^5$.

The following sum is a stochastic variable having a compound distribution

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

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

$$\mathrm{E}[W^n]=\sum _{i=0}^n\mathrm{E}\left[(\genfrac{}{}{0ex}{}{M}{i})\right]\sum _{j=0}^i(\genfrac{}{}{0ex}{}{i}{j}){(-1)}^{i-j}\mathrm{E}\left[{\left(\sum _{k=1}^j{Y}_k\right)}^n\right],$$

where $\mathrm{E}\left[{\left({\sum }_{k=1}^j{Y}_k\right)}^n\right]$ is defined as zero for $j=0$.

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 Template:Mvar-th moment, each term involving a value of Template:Mvar less than the mean by a certain amount exactly cancels out the term involving a value of Template:Mvar greater than the mean by the same amount.

Multivariate moments

For a continuous bivariate probability distribution with probability density function f(x,y) the (j,k) moment about the mean μ = (μ_X, μ_Y) is $$\begin{array}{rl}{\mu }_{j,k}& =\mathrm{E}\left[{(X-\mathrm{E}[X])}^j{(Y-\mathrm{E}[Y])}^k\right]\\ & =\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{(x-{\mu }_X)}^j{(y-{\mu }_Y)}^{k}f(x,y)dxdy.\end{array}$$

Central moment of complex random variables

The Template:Mvar-th central moment for a complex random variable Template:Mvar is defined as Template:Equation box 1 The absolute Template:Mvar-th central moment of Template:Mvar is defined as Template:Equation box 1

The 2nd-order central moment β₂ is called the variance of Template:Mvar whereas the 2nd-order central moment α₂ is the pseudo-variance of Template:Mvar.

See also

• Standardized moment
• Image moment
• Template:Slink
• Complex random variable