# Higher moments

## Introduction

### Moments

#### Moments

The $$n$$th moment of variable $$X$$ is defined as:

$$E[X^n]=\sum_i x_i^n P(x_i)$$

The mean is the first moment.

#### Central moments

The $$n$$th central moment of variable $$X$$ is defined as:

$$\mu_n=E[(X-E[X])^n]=\sum_i (x_i-E[X])^n P(x_i)$$

The variance is the second central moment.

#### Standardised moments

The $$n$$th standardised moment of variable $$X$$ is defined as:

$$\dfrac{E[(X-E[X])^n]}{(E[(X-E[X])^2]^\frac{n}{2}}=\dfrac{\mu_n}{\sigma^n}$$

#### Kertosis

Kertosis is the third standardised moment.

#### Skew

Skew is the fourth standardised moment.