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

Functional relationships for $T \sim WEIB(\mu,\sigma),\;\;T\in[0,\infty)$

$$ \begin{aligned} f(t|\mu,\sigma)&=\frac{1}{\sigma}\phi_{sev}\left(\frac{\log(t)-\mu}{\sigma}\right)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]\\\\ F(t|\mu,\sigma)&=\Phi_{sev}\left(\frac{\log(t)-\mu}{\sigma}\right)=1-\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right]\\\\ h(t|\mu,\sigma)&=\frac{1}{\sigma \exp(\mu)}\left[\frac{t}{\exp(\mu)}\right]^{1/\sigma-1}=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}\\\\ t_{p}&=\exp\left[\mu+\Phi^{-1}_{sev}(p)\sigma\right]=\eta[-log(1-p)]^{1/\beta}\\\\ E[T]&=\eta\Gamma(1+1/\beta)\\\\ Var[T]&=\eta^2\left[\Gamma(1+2/\beta)-\Gamma^2(1+1/\beta\right] \end{aligned} $$

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