In Auburngrads/teachingApps: Apps for Teaching Statistics, R Programming, and Shiny App Development

Functional relationships for $\;Y \sim NOR(\mu,\sigma),\;\;Y\in(-\infty,\infty)$

$$ \begin{aligned} f(y|\mu,\sigma)&=\frac{1}{\sigma}\phi_{nor}\left(\frac{y-\mu}{\sigma}\right)=\frac{1}{\sigma}\frac{e^{-(y - \mu)^{2}/(2\sigma^{2})}}{\sqrt{2\pi}}\\\\ F(y|\mu,\sigma)&=\Phi_{nor}\left(\frac{y-\mu}{\sigma}\right)=\int_{-\infty}^{y} \frac{e^{-(y-\sigma)^{2}/2\sigma^2}} {\sqrt{2\pi}\sigma}\\\\ h(y|\mu,\sigma)&=\frac{f(y|\mu,\sigma)}{1-F(y|\mu,\sigma)}\\\\ y_{p}&=\mu+\Phi^{-1}{nor}(p)\sigma, \;\;\;\;\;\;\;\;\text{where}\;\Phi^{-1}{nor}(p)=z_p\\\\ E[Y]&=\mu\\\\ Var[Y]&=\sigma^2 \end{aligned} $$

Auburngrads/teachingApps documentation built on June 17, 2020, 4:57 a.m.