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Wigner semicircle distribution

Wigner semicircle
Probability density function
Plot of the Wigner semicircle PDF
Cumulative distribution function
Plot of the Wigner semicircle CDF
Parameters radius (real)
Support
PDF
CDF
for
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

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 −RxR, 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 = 2RYR 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 Cn 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 into the defining equation for the moment generating function it can be seen that:


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