We recommend reading this documentation on pkgdown which renders math nicely.
https://pkg.mitchelloharawild.com/distributional/reference/dist_gev.html
In the following, let \(X\) be a GEV random variable with parameters
location = \(\mu\), scale = \(\sigma\), and shape = \(\xi\).
Support:
\(x \in \mathbb{R}\) (all real numbers) if \(\xi = 0\)
\(x \geq \mu - \sigma/\xi\) if \(\xi > 0\)
\(x \leq \mu - \sigma/\xi\) if \(\xi < 0\)
Mean:
$$
E(X) = \begin{cases}
\mu + \sigma \gamma & \text{if } \xi = 0 \\
\mu + \sigma \frac{\Gamma(1-\xi) - 1}{\xi} & \text{if } \xi < 1 \\
\infty & \text{if } \xi \geq 1
\end{cases}
$$
where \(\gamma \approx 0.5772\) is the Euler-Mascheroni constant and
\(\Gamma(\cdot)\) is the gamma function.
Median:
$$
\text{Median}(X) = \begin{cases}
\mu - \sigma \log(\log 2) & \text{if } \xi = 0 \\
\mu + \sigma \frac{(\log 2)^{-\xi} - 1}{\xi} & \text{if } \xi \neq 0
\end{cases}
$$
Variance:
$$
\text{Var}(X) = \begin{cases}
\frac{\pi^2 \sigma^2}{6} & \text{if } \xi = 0 \\
\frac{\sigma^2}{\xi^2} [\Gamma(1-2\xi) - \Gamma(1-\xi)^2] & \text{if } \xi < 0.5 \\
\infty & \text{if } \xi \geq 0.5
\end{cases}
$$
Probability density function (p.d.f):
For \(\xi = 0\) (Gumbel):
$$
f(x) = \frac{1}{\sigma} \exp\left(-\frac{x-\mu}{\sigma}\right)
\exp\left[-\exp\left(-\frac{x-\mu}{\sigma}\right)\right]
$$
For \(\xi \neq 0\):
$$
f(x) = \frac{1}{\sigma} \left[1 + \xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi-1}
\exp\left\{-\left[1 + \xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}
$$
where \(1 + \xi(x-\mu)/\sigma > 0\).
Cumulative distribution function (c.d.f):
For \(\xi = 0\) (Gumbel):
$$
F(x) = \exp\left[-\exp\left(-\frac{x-\mu}{\sigma}\right)\right]
$$
For \(\xi \neq 0\):
$$
F(x) = \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}
$$
where \(1 + \xi(x-\mu)/\sigma > 0\).
Quantile function:
For \(\xi = 0\) (Gumbel):
$$
Q(p) = \mu - \sigma \log(-\log p)
$$
For \(\xi \neq 0\):
$$
Q(p) = \mu + \frac{\sigma}{\xi}\left[(-\log p)^{-\xi} - 1\right]
$$
![[Stable]](figures/lifecycle-stable.svg?package=distributional&version=0.6.0)