Wigner semicircle distribution

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):

${\displaystyle f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,}$

Wigner semicircle

Probability density function
Cumulative distribution function
Parameters	${\displaystyle R>0\!}$ radius (real)
Support	${\displaystyle x\in [-R;+R]\!}$
PDF	${\displaystyle {\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!}$
CDF	${\displaystyle {\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!}$ for ${\displaystyle -R\leq x\leq R}$
Mean	${\displaystyle 0\,}$
Median	${\displaystyle 0\,}$
Mode	${\displaystyle 0\,}$
Variance	${\displaystyle {\frac {R^{2}}{4}}\!}$
Skewness	${\displaystyle 0\,}$
Ex. kurtosis	${\displaystyle -1\,}$
Entropy	${\displaystyle \ln(\pi R)-{\frac {1}{2}}\,}$
MGF	${\displaystyle 2\,{\frac {I_{1}(R\,t)}{R\,t}}}$
CF	${\displaystyle 2\,{\frac {J_{1}(R\,t)}{R\,t}}}$

for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution.

The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.

The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.