Wigner's semicircle rule, page 1

Wigner's semicircle rule says that the mean density of eigenvalues converges to

\psi(x) = \frac{1}{2\pi}\sqrt{4-x^2}

whenever the matrix entries are independent and satisfy some moment conditions.

Let's test this out with a bogus probability measure. Use the following cumulative distribution

I didn't bother to center at expectation 0 or scale to variance 1. But we should still get a (possibly shifted and stretched) semicircle.