# nLab Gaussian probability distribution

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Definition

A probability distribution on a Cartesian space $\mathbb{R}^n$ is called Gaussian or a normal distribution if it is of the form

$\mu_A \;\colon\; \vec x \mapsto \frac{\sqrt{det A}}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,d x^1 \cdots d x^n \,.$

where $A$ is some $n \times n$ matrix such that $\langle -, A-\rangle$ is a positive definite bilinear form. Here $\det A$ denotes the determinant and $\langle -,-\rangle$ is the canonical bilinear form on $\mathbb{R}^n$.

Since $\sqrt{\det A}$ is the coordinate of the volume element $vol_A$ associated with $A$, we may equivalently write this as

$\mu_A \;\colon\; \vec x \mapsto \frac{1}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,vol_A \,.$

The mean of this distribution is $\vec{0}$; for a distribution with mean $\vec{c}$, replace $\langle{\vec{x},A \vec{x}}\rangle$ with $\langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle$.

The matrix $A$ is the inverse of the covariance matrix?. In particular, for $n = 1$, we may write $x^2/\sigma^2$ (or $(x-c)^2/\sigma^2$ for mean $c$) in place of $\langle{\vec{x},A \vec{x}}\rangle$, where $\sigma$ is the standard deviation; similarly, $\sqrt{\det A}$ becomes $1/\sigma$. This gives the form

$\mu_\sigma \;\colon\; x \mapsto \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-c)^2}{2\sigma^2}\right) \,d x \,,$

which may be more familiar to some readers.