---
output: html_document
---

### Functional relationships for <script type='math/tex'>T \sim LOGLOGIS(\mu,\sigma),\;\;T\in(0,\infty)</script>

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