Functional relationships for

$$ \begin{aligned} f(y|\mu,\sigma)&=\frac{1}{\sigma}\phi_{logis}\left(\frac{y-\mu}{\sigma}\right)=\frac{1}{\sigma}\exp\left(\frac{y-\mu}{\sigma}\right)\left[1+\exp\left(\frac{y-\mu}{\sigma}\right)\right]^{-2}\\\\ F(y|\mu,\sigma)&=\Phi_{logis}\left(\frac{y-\mu}{\sigma}\right)=\exp\left(\frac{y-\mu}{\sigma}\right)\left[1+\exp\left(\frac{y-\mu}{\sigma}\right)\right]^{-1}\\\\ h(y|\mu,\sigma)&=\frac{1}{\sigma}\Phi_{logis}\left(\frac{y-\mu}{\sigma}\right)\\\\ y_{p}&=\mu+\Phi^{-1}{logis}(p)\sigma, \;\;\;\;\;\;\;\;\text{where}\;\Phi^{-1}{logis}(p)=\log [p/(1-p)]\\\\ E[Y]&=\mu\\\\ Var[Y]&=\sigma^2\pi^2/3 \end{aligned} $$



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