# Moment#

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

The $$n$$-th moment of a continuous function $$f(x)$$ of a real variable $$x$$ about a value $$c$$ is

$\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\diff x$

## Mean ($$n = 1$$)#

discrete:

$\mu_1 =\sum x_i \cdot P(x)$

with result $$x$$ and probability function $$P()$$

continuous:

$\operatorname \mu_1 = \int _{\mathbb {R}} x \cdot f(x)\diff x$

## Variance ($$n = 2$$)#

discrete: $$\(\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