Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / S / 1 / b / Solution

Draw independent uniforms $U_1,U_2$ on $(0,1)$, put $V=2U_1-1$, and accept $V$ if $U_2\le \sqrt{1-V^2}$; otherwise repeat independently. This is rejection sampling from the uniform proposal $q(v)=1/2$. Its accepted unnormalized density is $q(v)\sqrt{1-v^2}$, and the acceptance probability isDividing by $a$ gives the target Wigner semicircle distribution. The number of trials up to and including the first acceptance has a geometric distribution with success probability $\pi /4$, soThe expected number of rejected trials alone is $4/\pi -1$.