Table 1.

Common models representing a probability distribution.

Model	Equation
Normal distribution or Gaussian distribution	$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{{-\frac{1}{2}\left(\frac{x-\mu }{\sigma }\right)}^2}$
Pareto distribution	$F\left(x\right)=\left\{\begin{array}{cc}1-{\left(\frac{x_m}{x}\right)}^{\alpha },\hfill & x\ge x_m\hfill \\ 0,\hfill & x<x_m\hfill \end{array}\right)$
Continuous uniform distribution	$F\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x-\mu <-\sigma \sqrt{3}\hfill \\ \frac{1}{2}\left(\frac{x-\mu }{\sigma \sqrt{3}},+,1\right),\hfill & -\sigma \sqrt{3}\le x-\mu <\sigma \sqrt{3}\hfill \\ 1,\hfill & x-\mu \ge \sigma \sqrt{3}\hfill \end{array}\right)$
Weibull distribution	$F\left(N_f\right)=1-e^{-{\left(\frac{N_f}{\theta }\right)}^b}$
Gumbel distribution	$F\left(x,;,\mu ,,,\beta \right)=e^{-e^{-(x-\mu )/\beta }}$
Tukey lambda distribution	$Q\left(p,;,\lambda \right)=\left\{\begin{array}{cc}\frac{1}{\lambda }\left[p^{\lambda },-(1-p,{)}^{\lambda }\right],\hfill & \lambda \ne 0\hfill \\ log\left(\frac{p}{1-p}\right),\hfill & \lambda =0\hfill \end{array}\right)$
Exponential distribution	$f\left(x,,,\lambda \right)=\left\{\begin{array}{cc}\lambda e^{-\lambda x},\hfill & x\ge 0\hfill \\ 0,\hfill & x<0\hfill \end{array}\right)$