Functional relationships for

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



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