Wigner semicircle distribution

Wigner semicircle
Probability density function
Cumulative distribution function
Parameters	$R>0\!$ radius (real)
Support	$x \in [-R;+R]\!$
PDF	$\frac2{\pi R^2}\,\sqrt{R^2-x^2}\!$
CDF	$\frac12+\frac{x\sqrt{R^2-x^2}}{\pi R^2} + \frac{\arcsin\!\left(\frac{x}{R}\right)}{\pi}\!$ for $-R\leq x \leq R$
Mean	$0\,$
Median	$0\,$
Mode	$0\,$
Variance	$\frac{R^2}{4}\!$
Skewness	$0\,$
Ex. kurtosis	$-1\,$
Entropy	$\ln (\pi R) - \frac12 \,$
MGF	$2\,\frac{I_1(R\,t)}{R\,t}$
CF	$2\,\frac{J_1(R\,t)}{R\,t}$

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):

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

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.

General properties

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

E(X^{2n})=\left({R \over 2}\right)^{2n} C_n\,

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

C_n={1 \over n+1}{2n \choose n},\,

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

M(t)=\frac{2}{\pi}\int_0^\pi e^{Rt\cos(\theta)}\sin^2(\theta)\,d\theta

which can be solved (see Abramowitz and Stegun §9.6.18) to yield:

M(t)=2\,\frac{I_1(Rt)}{Rt}

where $I_1(z)$ is the modified Bessel function. Similarly, the characteristic function is given by:

\varphi(t)=2\,\frac{J_1(Rt)}{Rt}

where $J_1(z)$ is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving $\sin(Rt\cos(\theta))$ is zero.)

In the limit of $R$ approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.

Differential equation

$\left\{\left(r^{2}-x^{2}\right)f'(x)+xf(x)=0,f(1)={\frac {2{\sqrt {r^{2}-1}}}{\pi r^{2}}}\right\}$

Relation to free probability

In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely, in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.

References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

External links

Eric W. Weisstein et al., Wigner's semicircle

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Wigner semicircle distribution

General properties

Relation to free probability

See also

References

External links