Moment (mathematics)

Properties

Transformation of center

Since $${\displaystyle (x-b{)}^n=(x-a+a-b{)}^n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})(x-a{)}^i(a-b{)}^{n-i}}$$ where $(\genfrac{}{}{0ex}{}{n}{i})$ is the binomial coefficient, it follows that the moments about b can be calculated from the moments about a by: $${\displaystyle E\left[(x-b{)}^n\right]=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})E\left[(x-a{)}^i\right](a-b{)}^{n-i}.}$$

As a corollary of this result, the lowest nonzero moment is independent of the choice of center. We can see this from the formula above by considering the case where the ith moments (about some point a) are zero for all ${\displaystyle i<n}$.

Moment of a convolution of function

The raw moment of a convolution $h(t)=(f\ast g)(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )g(t-\tau )d\tau$ reads $${\displaystyle {\mu }_n[h]=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i}){\mu }_i[f]{\mu }_{n-i}[g]}$$ where ${\displaystyle {\mu }_n[\cdot ]}$ denotes the ${\displaystyle n}$th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain rule for differentiating a product.

Partial moments

Partial moments are sometimes referred to as "one-sided moments". The nth order lower and upper partial moments with respect to a reference point r may be expressed as $${\displaystyle {\mu }_n^{-}(r)={\int }_{-\mathrm{\infty }}^r(r-x{)}^{n}f(x)\mathrm{d}x,}$$ $${\displaystyle {\mu }_n^{+}(r)={\int }_r^{\mathrm{\infty }}(x-r{)}^{n}f(x)\mathrm{d}x.}$$

If the integral function does not converge, the partial moment does not exist.

Partial moments are normalized by being raised to the power 1/n. The upside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment.

Central moments in metric spaces

Let (M, d) be a metric space, and let B(M) be the Borel σ-algebra on M, the σ-algebra generated by the d-open subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞.

The pth central moment of a measure μ on the measurable space (M, B(M)) about a given point x₀ ∈ M is defined to be $${\displaystyle {\int }_{M}d{\left(x,x_0\right)}^p\mathrm{d}\mu (x).}$$

μ is said to have finite pth central moment if the pth central moment of μ about x₀ is finite for some x₀ ∈ M.

This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, P) is a probability space and X : Ω → M is a random variable, then the pth central moment of X about x₀ ∈ M is defined to be $${\displaystyle {\int }_{M}d{\left(x,x_0\right)}^p\mathrm{d}\left(X_{\ast }\left(\mathbf{P}\right)\right)(x)={\int }_{\mathrm{\Omega }}d{\left(X(\omega ),x_0\right)}^p\mathrm{d}\mathbf{P}(\omega )=\mathbf{E}[d(X,x_0{)}^p],}$$ and X has finite pth central moment if the pth central moment of X about x₀ is finite for some x₀ ∈ M.