Wigner semicircle distribution

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	${\displaystyle 2\,{\frac {J_{1}(R\,t)}{R\,t}}}$

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse):

for −R ≤ x ≤ R, and f(x) = 0 if R < |x|.

This distribution arises as the limiting distribution of eigenvalues of many random symmetric matrices as the size of the matrix approaches infinity.

It is a scaled beta distribution, more precisely, if Y is beta distributed with parameters α = β = 3/2, then X = 2RY – R has the above Wigner semicircle distribution.

The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution.

For positive integers n, the 2n-th moment of this distribution is

where X is any random variable with this distribution and C_n is the nth Catalan number

so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)

Making the substitution ${\displaystyle x=R\cos(\theta )}$ into the defining equation for the moment generating function it can be seen that: