Stunts with Gaussian distributions.
Let’s start here with the basic thing. The (univariate) standard Gaussian pdf
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.
Non-linear univariate DE represention.
Linear PDE representation as a diffusion equation
(see, e.g. BoGK10)
Look, it’s the diffusion equation of Wiener process. Surprise.
As made famous by Wiener processes in finance and Gaussian processes in Bayesian nonparametrics.
See, e.g. these lectures.
- 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.
- Botev, Z. I., Grotowski, J. F., & Kroese, D. P.(2010) Kernel density estimation via diffusion. The Annals of Statistics, 38(5), 2916–2957. DOI.