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 2"n"-th moment of this distribution is
formula_12
where "X" is any random variable with this distribution and "C""n" is the "n"th Catalan number
formula_13
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 formula_14 into the defining equation for the moment generating function it can be seen that:
formula_15
which can be solved (see Abramowitz and Stegun §9.6.18)
to yield:
formula_16
where formula_17 is the modified Bessel function. Similarly, the characteristic function is given by:
formula_18
where formula_19 is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving formula_20 is zero.) In the limit of formula_21 approaching zero, the Wigner semicircle distribution becomes a Dirac delta function. 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. The parabolic probability distribution supported on the interval [−"R", "R"] of radius "R" centered at (0, 0):
formula_22
for −"R" ≤ "x" ≤ "R", and "f"("x") = 0 if "|x|" > "R".
Example. The joint distribution is
formula_23
formula_24
Hence, the marginal PDF of the spherical (parametric) distribution is:
formula_25
formula_26
formula_27 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. The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0):
formula_28,
for −1 ≤ "x" ≤ 1, and "f"("x") = 0 if "|x|" > 1.
Example. The joint distribution is
formula_29
formula_24
Hence, the marginal PDF distribution is 
formula_31 such that R=1
The cumulative distribution function (CDF) is
formula_32 such that R=1 and n >= -1
The characteristic function (CF) of the PDF is related to the beta distribution as shown below
formula_33
In terms of Bessel functions this is
formula_34
Raw moments of the PDF are
formula_35
Central moments are
formula_36
formula_37
formula_38
formula_39
formula_40
The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are: formula_41
formula_42
formula_43
formula_44
formula_45
Raw moments of the characteristic function are:
formula_46
For an even distribution the moments are 
formula_47
formula_48
formula_49
formula_50
formula_51
formula_52
formula_53
formula_54
formula_55
formula_56
formula_57
formula_58
Hence, the moments of the CF (provided N=1) are
formula_59
formula_60
formula_61
formula_62
formula_63
Skew and Kurtosis can also be simplified in terms of Bessel functions. The entropy is calculated as
formula_64
The first 5 moments (n=-1 to 3), such that R=1 are
formula_65
formula_66
formula_67
formula_68
formula_69 The marginal PDF distribution with odd symmetry is 
formula_70 such that R=1
Hence, the CF is expressed in terms of Struve functions
formula_71
"The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" 
formula_72
formula_73
<math> X_1 =
General properties
α = β = 3/2, then "X" = 2"RY" – "R" has the Wigner semicircle distribution.