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

Functional relationships for $Y \sim SEV(\mu,\sigma),\;\;Y\in(-\infty,\infty)$

$$ \begin{aligned} f(y|\mu,\sigma)&=\frac{1}{\sigma}\phi_{sev}\left(\frac{y-\mu}{\sigma}\right)=\frac{1}{\sigma}e^{\left(\frac{t-\mu}{\sigma}\right)}e^{-e^{\left(\frac{t-\mu}{\sigma}\right)}}\\\\ F(y|\mu,\sigma)&=\Phi_{sev}\left(\frac{y-\mu}{\sigma}\right)=1-e^{-e^{\left(\frac{y-\mu}{\sigma}\right)}}\\\\ h(y|\mu,\sigma)&=\frac{1}{\sigma}\exp\left(\frac{y-\mu}{\sigma}\right)\\\\ y_{p}&=\mu+\Phi^{-1}{sev}(p)\sigma, \;\;\;\;\;\;\text{where}\;\Phi^{-1}{sev}(p)=\log[-\log(1-p)]\\\\ E[Y]&=\mu-\sigma\gamma, \;\;\;\;\;\;\;\;\;\;\;\; \text{where}\;\;\gamma\approx 0.5772\;\;\text{(Euler's Constant)}\\\\ Var[Y]&=\sigma^2\pi^2/6 \end{aligned} $$

teachingApps documentation built on July 1, 2020, 5:58 p.m.