We recommend reading this documentation on pkgdown which renders math nicely.
https://pkg.mitchelloharawild.com/distributional/reference/dist_gpd.html
In the following, let \(X\) be a Generalized Pareto random variable with
parameters location = \(a\), scale = \(b > 0\), and
shape = \(s\).
Support:
\(x \ge a\) if \(s \ge 0\),
\(a \le x \le a - b/s\) if \(s < 0\)
Mean:
$$
E(X) = a + \frac{b}{1 - s} \quad \textrm{for } s < 1
$$
\(E(X) = \infty\) for \(s \ge 1\)
Variance:
$$
\textrm{Var}(X) = \frac{b^2}{(1-s)^2(1-2s)} \quad \textrm{for } s < 0.5
$$
\(\textrm{Var}(X) = \infty\) for \(s \ge 0.5\)
Probability density function (p.d.f):
For \(s = 0\):
$$
f(x) = \frac{1}{b}\exp\left(-\frac{x-a}{b}\right) \quad \textrm{for } x \ge a
$$
For \(s \ne 0\):
$$
f(x) = \frac{1}{b}\left(1 + s\frac{x-a}{b}\right)^{-1/s - 1}
$$
where \(1 + s(x-a)/b > 0\)
Cumulative distribution function (c.d.f):
For \(s = 0\):
$$
F(x) = 1 - \exp\left(-\frac{x-a}{b}\right) \quad \textrm{for } x \ge a
$$
For \(s \ne 0\):
$$
F(x) = 1 - \left(1 + s\frac{x-a}{b}\right)^{-1/s}
$$
where \(1 + s(x-a)/b > 0\)
Quantile function:
For \(s = 0\):
$$
Q(p) = a - b\log(1-p)
$$
For \(s \ne 0\):
$$
Q(p) = a + \frac{b}{s}\left[(1-p)^{-s} - 1\right]
$$
Median:
For \(s = 0\):
$$
\textrm{Median}(X) = a + b\log(2)
$$
For \(s \ne 0\):
$$
\textrm{Median}(X) = a + \frac{b}{s}\left(2^s - 1\right)
$$
Skewness and Kurtosis: No closed-form expressions; approximated numerically.