Distributions

parameters

PDF

CDF

genpareto

$\mu \in \left(-\infty ,\infty \right)$ location

$\sigma \in \left(0,\infty \right)$ scale

$\xi \in \left(-\infty ,\infty \right)$ shape

$\frac{1}{\sigma }{\left(1+\xi z\right)}^{-\left(1/\xi +1\right)}$

Where $z=\frac{x-\mu }{\sigma }$

$1-{\left(1+\xi z\right)}^{-\frac{1}{\xi }}$

Exponential

$\lambda >0$ , rate, or inverse scale

$\lambda {\text{e}}^{-\lambda x}$

$1-{\text{e}}^{-\lambda x}$

Weibull

$\lambda \in \left(0,+\infty \right)$ scale

$k\in \left(0,+\infty \right)$ shape

$f\left(x\right)=\{\begin{array}{l}\frac{k}{\lambda }{\left(\frac{x}{\lambda }\right)}^{k-1}{\text{e}}^{-{\left(x/\lambda \right)}^k}x\ge 0\\ 0x<0\end{array}$

$\{\begin{array}{l}1-{\text{e}}^{-{\left(x/\lambda \right)}^k}x\ge 0\\ 0x<0\end{array}$

Gumbel

$\mu$ location

$\beta >0$ scale

$\frac{1}{\beta }{\text{e}}^{-\text{e}\left(z+{\text{e}}^{-x}\right)}$

Where $z=\frac{x-\mu }{\beta }$

${\text{e}}^{-\text{e}\left(x-\mu \right)/\beta }$

Log-normal

$\mu \in \left(-\infty ,+\infty \right)$

$\sigma >0$

$\frac{1}{x\sigma \sqrt{2\pi }}\exp \left(-\frac{{\left(\ln x-\mu \right)}^2}{2{\sigma }^2}\right)$

$\frac{1}{2}+\frac{1}{2}\text{erf}\left(\frac{\ln x-\mu }{\sqrt{2}\sigma }\right)$

Gamma

$k>0$ scale

$\theta >0$ shape

$\frac{1}{\Gamma \left(k\right){\theta }^k}{x}^{k-1}{\text{e}}^{-\frac{x}{\theta }}$

$\frac{1}{\Gamma \left(k\right)}\gamma \left(k,\frac{x}{\theta }\right)$

Fréchet

$s\in \left(0,\infty \right)$ scale

$m\in \left(0,\infty \right)$ location

$\frac{\text{α}}{s}{\left(\frac{x-m}{s}\right)}^{-1-\alpha }{\text{e}}^{-{\left(\frac{x-m}{s}\right)}^{-\alpha }}$

${\text{e}}^{-{\left(\frac{x-m}{s}\right)}^{-\alpha }}$