\[X\sim{\rm Beta-Prime}(\alpha,\beta)\]
\[f_X(x|\alpha,\beta) =\frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\]
\[F_X(x|\alpha,\beta) = I_{\frac{x}{1+x}}\left(\alpha, \beta \right),\] where ‘’I’’ is the regularized incomplete beta function.
\[X\sim{\rm \chi^{2}_{k}}\]
\[f_X(x|k) = \frac{1}{2^{k/2}\Gamma(k/2)}\; x^{k/2-1} e^{-x/2}\]
\[F_X(x|k) = \frac{1}{\Gamma(k/2 )} \; \gamma\left(\frac{k}{2},\,\frac{x}{2}\right)\]
\[X \sim{\rm Dagum}(a,b,p)\]
\[f_X(x|a,b,p) = \frac{ap}{x} \left( \frac{(\tfrac{x}{b})^{a p}}{\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)\] \[F_X(x|a,b,p) = \left( 1+\left(\frac{x}{b}\right)^{-a} \right)^{-p} \text{ for } x > 0 \text{ where } a, b, p > 0 .\]
\[X\sim{\rm Exponential}(\lambda)\]
\[f_X(x|\lambda)=\lambda e^{-\lambda x}\] \[F_X(x|\lambda)=1- e^{-\lambda x}\]
\[X\sim{\rm F}(d_{1},d_{2})\]
\[f_X(x|d_{1},d_{2}) = \frac{1}{\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2} - 1} \left(1+\frac{d_1}{d_2}\,x\right)^{-\frac{d_1+d_2}{2}}\]
\[F_X(x|d_{1},d_{2})= I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,\] where ‘’I’’ is the regularized incomplete beta function.
\[X\sim{\rm Gamma}(\alpha,\beta)\]
\[f_X(x|\alpha,\beta)= \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x }\] \[F_X(x|\alpha,\beta) = \frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x)\]
\[X\sim{\rm IG}(\mu,\lambda)\] \[f_X(x|\mu, \lambda) = \sqrt\frac{\lambda}{2 \pi x^3} \exp\left[-\frac{\lambda (x-\mu)^2}{2 \mu^2 x}\right]\]
\[F_X(x|\mu, \lambda)= \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right)\right) +\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1 \right)\right)\]
\[X\sim{\rm log-Cauchy}(\mu,\sigma)\] \[f_X(x|\mu,\sigma)={ 1 \over x\pi } \left[ { \sigma \over (\ln x - \mu)^2 + \sigma^2 } \right], \ \ x>0\] \[F_X(x|\mu,\sigma)=\frac{1}{\pi} \arctan\left(\frac{\ln x-\mu}{\sigma}\right)+\frac{1}{2}, \ \ x>0\]
\[X\sim{\rm log-logistic}(\alpha,\beta)\]
\[f_X(x|\alpha,\beta)=\frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} } { \left( 1+(x/\alpha)^{\beta} \right)^2 }\]
\[F_X(x|\alpha,\beta)= {x^\beta \over \alpha^\beta+x^\beta}\]
\[\ln(X)\sim{\rm \mathcal N}(\mu,\sigma)\] \[f_X(x|\mu,\sigma) = \frac 1 {x\sigma\sqrt{2\pi}}\ \exp\left(-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}\right)\]
\[F_X(x|\mu,\sigma)= \frac12 + \frac12\operatorname{erf}\Big[\frac{\ln x-\mu}{\sqrt{2}\sigma}\Big]\]
\[X\sim{\rm Pareto}(x_{m}, \alpha)\]
\[f_X(x|x_{m}, \alpha) = \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}\]
\[F_X(x|x_{m}, \alpha)=1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha\]
\[X\sim{\rm Weibull}(\lambda,k)\]
\[f_X(x|\lambda,k)=\begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k} & x\geq0\\ 0 & x<0\end{cases}\]
\[F_X(x|\lambda,k)=\begin{cases}1- e^{-(x/\lambda)^k} & x\geq0\\ 0 & x<0\end{cases}\]