Distribution

Pdf and cdf

Range

Moments

Normal

$f\left(x\right)=\frac{1}{{\sigma }_x\sqrt{2\pi }}\exp \left[-\frac{1}{2}{\left(\frac{x-{\mu }_x}{{\sigma }_x}\right)}^2\right]$

$-\infty <x<\infty$

${\mu }_x$ and ${\sigma }_x^2$ ; ${\gamma }_x=0$

Lognormal

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

$x>0$

${\mu }_x=\exp \left({\mu }_y+\frac{{\sigma }_y^2}{2}\right)$ , ${\sigma }_x^2={\mu }_x^2\left[\exp \left({\sigma }_y^2\right)-1\right]$ ; ${\gamma }_x=3C{V}_x+C{V}_x^3$

Pearson type 3

$f\left(x\right)=\frac{{\lambda }^{\beta }{\left(x-ϵ\right)}^{\beta -1}{\text{e}}^{-\lambda \left(x-ϵ\right)}}{\Gamma \left(\beta \right)}$

$\beta >0$

$\beta ={\left(\frac{2}{{\gamma }_x}\right)}^2$ , $\lambda =\frac{\sqrt{\beta }}{{\sigma }_x}$ and $ϵ={\mu }_x-\frac{\beta }{\lambda }$

Gumbel

$f\left(x\right)=\frac{1}{\alpha }\exp \left[-\frac{x-\xi }{\alpha }-\exp \left(-\frac{x-\xi }{\alpha }\right)\right]$

$F\left(x\right)=\exp \left[-\exp \left(-\frac{x-\xi }{\alpha }\right)\right]$

$-\infty <x<\infty$

${\mu }_x=\xi +0.5772\alpha$

${\sigma }_x^2=\frac{{\pi }^2{\alpha }^2}{6}\approx 1.645{\alpha }^2$ ; ${\gamma }_x=1.1396$