Stunts with Gaussian distributions.
Let’s start here with the basic thing. The (univariate) standard Gaussian pdf
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.
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.
As made famous by Wiener processes in finance and Gaussian processes in Bayesian nonparametrics.
- Botev, Z. I.(2017) The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(1), 125–148. 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.
- Steinbrecher, G., & Shaw, W. T.(2008) Quantile mechanics. European Journal of Applied Mathematics, 19(2), 87–112. DOI.
- Wichura, M. J.(1988) Algorithm AS 241: The Percentage Points of the Normal Distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 37(3), 477–484. DOI.