Wigner semicircle distribution

The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval 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): The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval 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 |x| > R. 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. A higher-dimensional generalization is a parabolic distribution in three dimensional space, namely the marginal distribution function of a spherical (parametric) distribution f X , Y , Z ( x , y , z ) = 3 4 π , x 2 + y 2 + z 2 ≤ 1 , {displaystyle f_{X,Y,Z}(x,y,z)={frac {3}{4pi }},qquad qquad x^{2}+y^{2}+z^{2}leq 1,} f X ( x ) = ∫ − 1 − y 2 − x 2 + 1 − y 2 − x 2 ∫ − 1 − x 2 + 1 − x 2 3 d y 4 π = 3 ( 1 − x 2 ) / 4. {displaystyle f_{X}(x)=int _{-{sqrt {1-y^{2}-x^{2}}}}^{+{sqrt {1-y^{2}-x^{2}}}}int _{-{sqrt {1-x^{2}}}}^{+{sqrt {1-x^{2}}}}{frac {3mathrm {d} y}{4pi }}=3(1-x^{2})/4.}