R] of radius R centered at (0, 0): for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Example. The joint distribution is Hence, the marginal PDF of the spherical (parametric) distribution is: such that R=1 The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z. The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals. Wigner n-sphere distribution The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0): , for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1. Example. The joint distribution is Hence, the marginal PDF distribution is such that R=1 The cumulative distribution function (CDF) is such that R=1 and n >= -1 The characteristic function (CF) of the PDF is related to the beta distribution as shown below In terms of Bessel functions this is Raw moments of the PDF are Central moments are The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are: Raw moments of the characteristic function are: For an even distribution the moments are Hence, the moments of the CF (provided N=1) are Skew and Kurtosis can also be simplified in terms of Bessel functions. The entropy is calculated as The first 5 moments (n=-1 to 3), such that R=1 are N-sphere Wigner distribution with odd symmetry applied The marginal PDF distribution with odd symmetry is such that R=1 Hence, the CF is expressed in terms of Struve functions "The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" Example (Normalized Received Signal Strength): quadrature terms The normalized received signal strength is defined as and using standard quadrature terms Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining The expanded form of the Characteristic function of the received signal strength becomes See also Wigner surmise The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity. In number-theoretic literature, the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture. Marchenko–Pastur distribution or Free Poisson distribution References Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.