# Moment 

##### Definition

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

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

## Mean ($n = 1$)

discrete:

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

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

continuous:

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

## 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