$$ \begin{aligned} f(t|\mu,\sigma)&=\frac{1}{\sigma}\phi_{nor}\left(\frac{\log(t)-\mu}{\sigma}\right)\\\\ F(t|\mu,\sigma)&=\Phi_{nor}\left(\frac{\log(t)-\mu}{\sigma}\right)\\\\ h(t|\mu,\sigma)&=\frac{f(t|\mu,\sigma)}{1-F(t|\mu,\sigma)}\\\\ t_{p}&=\exp\left[\mu+\Phi^{-1}{nor}(p)\sigma\right], \;\;\;\;\;\;\;\;\text{where}\;\Phi^{-1}{nor}(p)=z_p\\\\ E[T]&=\exp(\mu+0.5\sigma^2)\\\\ Var[T]&=\exp(2\mu+\sigma^2)(\exp(\sigma^2)-1) \end{aligned} $$
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