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### Functional relationships for <script type='math/tex'>Y \sim LEV(\mu,\sigma),\;\;Y\in(-\infty,\infty)</script>

$$
\begin{aligned}
f(t|\mu,\sigma)&=\frac{1}{\sigma}\phi_{lev}\left(\frac{y-\mu}{\sigma}\right)=\frac{1}{\sigma}e^{\left(-\frac{y-\mu}{\sigma}\right)}e^{-e^{\left(-\frac{y-\mu}{\sigma}\right)}}\\\\\\\\
F(t|\mu,\sigma)&=\Phi_{lev}\left(\frac{y-\mu}{\sigma}\right)=e^{-e^{\left(-\frac{y-\mu}{\sigma}\right)}}\\\\\\\\
h(t|\mu,\sigma)&=\frac{\exp\left(-\frac{y-\mu}{\sigma}\right)}{\sigma \left(\exp\left[\exp\left(-\frac{y-\mu}{\sigma}\right)\right]-1\right)}\\\\\\\\
t_{p}&=\exp\left[\mu+\Phi^{-1}_{sev}(p)\sigma\right]\\\\\\\\
E[T]&=\mu+\sigma\gamma\\\\\\\\
Var[T]&=\sigma^2\pi^2/6
\end{aligned}
$$