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Definition

A moment is a specific quantitative measure of the shape of a function.

The nn-th moment of a continuous function f(x)f(x) of a real variable xx about a value cc is

μn=(xc)nf(x)dx\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\,\text{d}x

Mean (n=1n = 1)

discrete:

μ1=xiP(x)\mu_1 =\sum x_i \cdot P(x)

with result xx and probability function P()P()

continuous:

μ1=xf(x)dx\mathop{\mu}_1 = \int _{\mathbb {R}} x \cdot f(x)\,\text{d}x

Variance (n=2n = 2)

discrete: μ2=xP(x)\mu_2 =\sum x \cdot P(x)

Central Moments Comparison

Moment Statistics Mechanics
0. Total P=1.0 mass
1. mean center of mass
2. variance rotational inertia
3. skewness