The Living Thing / Notebooks :

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 , 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

Rational approximations

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.

Roughness

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.

Transformed variables

implies

Refs