Johnson SB Distribution¶

There are two shape parameters \(a\in\mathbb{R}\) and \(b>0\), and the support is \(x\in\left[0,1\right]\).

\begin{eqnarray*} f\left(x;a,b\right) & = & \frac{b}{x\left(1-x\right)}\phi\left(a+b\log\frac{x}{1-x}\right)\\ F\left(x;a,b\right) & = & \Phi\left(a+b\log\frac{x}{1-x}\right)\\ G\left(q;a,b\right) & = & \frac{1}{1+\exp\left(-\frac{1}{b}\left(\Phi^{-1}\left(q\right)-a\right)\right)}\end{eqnarray*}