Gaussian distribution and Erf and Normality

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Stunts with Gaussian distributions.

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

\begin{equation*}
\psi:x\mapsto \frac{1}{sqrt{2\pi}}\text{exp}\left(-\frac{x^2}{2}\right)
\end{equation*}

We define
.. math:

\Psi:x\mapsto \int_{-\infty}^x\psi{t} dt

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

Well…

\begin{equation*}
\operatorname{erf}(x)\; =\; \frac{1}{\sqrt{\pi}} \int_{-x}^x e^{-t^2} \, dt
\end{equation*}
\begin{equation*}
\sqrt{\frac{\pi }{2}} \left(\text{erf}\left(\frac{x}{\sqrt{2}}\right)+1\right)
\end{equation*}

Differential representation

Non-linear univariate DE represention.

\begin{equation*}
\begin{align*}
\sigma ^2 f'(x)+f(x) (x-\mu )&=0\\
f(0) &=\frac{e^{-\mu ^2/(2\sigma ^2)}}{\sqrt{2 \sigma^2\pi } }\\
L(x) &=(\sigma^2 D+x-\mu)
\end{align*}
\end{equation*}

Linear PDE representation as a diffusion equation (see, e.g. BoGK10)

\begin{equation*}
\begin{align*}
\frac{\partial}{\partial t)f(x;t) &=\frac{1}{2}\frac{\partial^2}{\partial x^2}f(x;t)\\
f(x;0)&=\delta(x-\mu)
\end{align*}
\end{equation*}

Look, it’s the diffusion equation of Wiener process.

Roughness

\begin{equation*}
\begin{align*}
\| \frac{d}{dx}\phi_\sigma \|_2 &= \frac{1}{4\sqrt{\pi}\simga^3}\\
\| \left(\frac{d}{dx}\right)^n \phi_\sigma \|_2 &= \frac{\prod_{i<n}2n-1}{2^{n+1}\sqrt{\pi}\simga^{2n+1}}
\end{align*}
\end{equation*}

Refs

Bote16
Botev, Z. I.(2016) The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), n/a-n/a. DOI.
BoGK10
Botev, Z. I., Grotowski, J. F., & Kroese, D. P.(2010) Kernel density estimation via diffusion. The Annals of Statistics, 38(5), 2916–2957. DOI.

See original: The Living Thing / Notebooks Gaussian distribution and Erf and Normality