# 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.