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

Functional relationships for $\;T \sim BISA(\theta,\beta),\;\;t > 0;\;\;\theta\;>0,\;\;\beta\;>0$

$$ \begin{aligned} f(t|\theta,\beta)&=\frac{\sqrt{\frac{t}{\theta}}+\sqrt{\frac{\theta}{t}}}{2\beta{t}}\phi_{nor}(z),\;\;\;\;where\;\;z=\frac{1}{\beta}\left(\sqrt{\frac{t}{\theta}}-\sqrt{\frac{\theta}{t}}\right)\\\\ F(t|\theta,\beta)&=\Phi_{nor}(z)\\\\ h(t)&=\frac{\sqrt{\frac{t}{\theta}}-\sqrt{\frac{\theta}{t}}}{2\beta{t}}\left[\frac{\phi(z)}{\Phi(-z)}\right]\\\\ t_p&=\frac{\theta}{4}\left{\beta\Phi_{nor}^{-1}(p)+\sqrt{4+[\beta\Phi_{nor}^{-1}(p)]^{2}}\right}^{2}\\\\ E[T]&=\theta\left(1+\frac{\beta^{2}}{2}\right)\\\\ Var[T]&=(\theta\beta)^{2}\left(1+\frac{5\beta^{2}}{4}\right) \end{aligned} $$

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