= Wigner semicircle distribution
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{title2=$f(x)=\frac2\pi\sqrt{1-x^2}\,\mathbf1_{(-1,1)}(x)$}
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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>.
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