Arcsine distribution ($\mathrm{Beta}(\frac{1}{2},\frac{1}{2})$)

The standard arcsine law has probability density function $1/[\pi \sqrt{x(1-x)}]$ on $(0,1)$. Substituting $x={\sin }^{2}v$ in its integral gives cumulative distribution function $F(x)=2\arcsin \sqrt{x}/\pi$. Hence inverse transform sampling uses $X={\sin }^2(\pi U/2)$ for $U$ with a continuous uniform distribution. The law places most of its density near the two endpoints rather than the middle.