# The Gaussian distribution

### and Erf and normality and such

Stunts with Gaussian distributions.

Let's start here with the basic thing. The (univariate) standard Gaussian pdf

We define

## Left tail of icdf

For small $p$, the quantile function has the useful asymptotic expansion

## What is Erf again?

This erf function is popular, isn't it? Unavoidable if you do computer algebra. But I can never remember what it is. There's these two scaling factors tacked on.

Well…

and

Done.

## Representations

TBD.

### ODE representation for the univariate density

TODO: note where I learned this.

### ODE representation for the icdf

From StSh08 via Wikipedia.

### Density PDE representation as a diffusion equation

(see, e.g. BoGK10)

Look, it's the diffusion equation of Wiener process. Surprise.

Univariate -

## Multidimensional marginals

As made famous by Wiener processes in finance and Gaussian processes in Bayesian nonparametrics.

See, e.g. these lectures, or Michael I Jordan's backgrounders.

implies