Semicircular Distribution¶

Defined on \(x\in\left[-1,1\right]\)

\begin{eqnarray*} f\left(x\right) & = & \frac{2}{\pi}\sqrt{1-x^{2}}\\ F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\left[x\sqrt{1-x^{2}}+\arcsin x\right]\\ G\left(q\right) & = & F^{-1}\left(q\right)\end{eqnarray*}

\begin{eqnarray*} m_{d}=m_{n}=\mu & = & 0\\ \mu_{2} & = & \frac{1}{4}\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & -1\end{eqnarray*}