Function

Formula

Beta


f(x)=\frac{1}{B(\alpha,\beta)}\frac{(x-lower)^{\alpha-1}(upper-x)^{\beta-1}}{(upper-lower)^{\alpha+\beta-1}}

B(\alpha,\beta) is the Beta function

Exponential

f(x)=\frac{1}{\mu}\exp(-\frac{x-shift}{\mu})

Gamma



f(x)=\frac{(x-shift)^{\alpha-1}}{\beta^{\alpha}}\frac{1}{\Gamma(\alpha)}\exp(-\frac{x-min}{\beta})

Shape=α, Scale=β

\Gamma(\alpha) is the Gamma function

Generalized Extreme Value

(Simple Distribution only)

f(x)=\frac{1}{\alpha}e^{-(1-\kappa)y-e^{-y}}

y=\frac{-1}{\kappa}log\{1-\kappa\frac{(x-\xi)}{\alpha}\}, \kappa \neq 0

y=(x-\xi)/\alpha, \kappa = 0

Location = ξ, Scale = α, Shape = κ

Gumbel

f(x)=\frac{1}{\alpha}\exp(-(\frac{x-\xi}{\alpha}+\exp(-\frac{x-\xi}{\alpha})))

Location = ξ, Scale = α

Kappa

(Simple Distribution only)

f(x)=\frac{1}{\alpha}[1-\kappa\frac{(x-\xi)}{\alpha}]^{(1/\kappa)-1}[F(x)]^{1-h}

F(x)=\{1-h[1-\kappa\frac{(x-\xi)}{\alpha}]^{1/\kappa}\}^{1/h}

F(x) is the CDF

Location = ξ, Scale = α, Shape 1 = κ, Shape 2 = h

Log-normal

f(x)=\frac{1}{x\sqrt{2\pi\sigma^{2}}}\exp(-\frac{(\log(x)-\mu)^{2}}{2\sigma^{2}})

Normal

f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}})

Triangular


f(x)=\begin{cases} 0, & x < a,\\ \frac{2(x-a)}{(b-a)(c-a)}, & a \le x < c,\\ \frac{2}{b-a}, & x = c,\\ \frac{2(b-x)}{(b-a)(b-c)}, & c < x \le b,\\ 0, & b < x. \end{cases}

Lower = a, Upper = b, Mode = c

Uniform


f(x)=\begin{cases} \frac{1}{b-a}, & x \in [a,b] \\ 0, & \text{otherwise}. \end{cases}

Lower=a, Upper=b

Weibull


f(x)=\begin{cases} \frac{k}{\lambda}(\frac{x-shift}{\lambda})^{k-1}\exp(-(\frac{x-shift}{\lambda})^{k}), & x \ge 0 \\ 0, & x < 0 \end{cases}

Shape=k, Scale=λ