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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 $4/\pi -1$.

The unit-radius Wigner semicircle distribution has the displayed probability density function. Its graph is a rescaled upper semicircle. The area of the unit semicircle is $\pi /2$, so the factor $2/\pi$ normalizes it. Symmetry gives mean zero, and the substitution $x=\sin \theta$ gives variance $1/4$.

A scaled radius-$R$ version has density $2\sqrt{R^2-x^2}/(\pi R^2)$ on $(-R,R)$ and variance $R^2/4$. Rejection sampling from a uniform proposal on $(-1,1)$ accepts a proposed $x$ with probability $\sqrt{1-x^2}$; its average acceptance probability is $\pi /4$. Thus the number of proposals for one output has geometric distribution with expectation $4/\pi$. This law also describes the limiting empirical eigenvalue density in standard ensembles of large symmetric random matrices.