We recommend reading this documentation on
https://alexpghayes.github.io/distributions3/, where the math
will render with additional detail and much greater clarity.
In the following, let \(X\) be a Frechet random variable with location
parameter location = \(m\), scale parameter scale =
\(s\), and shape parameter shape = \(\alpha\).
A Frechet(\(m, s, \alpha\)) distribution is equivalent to a
\link{GEV}(\(m + s, s / \alpha, 1 / \alpha\)) distribution.
Support: \((m, \infty)\).
Mean: \(m + s\Gamma(1 - 1/\alpha)\), for \(\alpha > 1\); undefined
otherwise.
Median: \(m + s(\ln 2)^{-1/\alpha}\).
Variance:
\(s^2 [\Gamma(1 - 2 / \alpha) - \Gamma(1 - 1 / \alpha)^2]\)
for \(\alpha > 2\); undefined otherwise.
Probability density function (p.d.f):
$$f(x) = \alpha s ^ {-1} [(x - m) / s] ^ {-(1 + \alpha)}%
\exp\{-[(x - m) / s] ^ {-\alpha} \}$$
for \(x > m\). The p.d.f. is 0 for \(x \leq m\).
Cumulative distribution function (c.d.f):
$$F(x) = \exp\{-[(x - m) / s] ^ {-\alpha} \}$$
for \(x > m\). The c.d.f. is 0 for \(x \leq m\).