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			name =Wigner semicircle\|
			type =density\|
			pdf_image =[[Image:WignerS_distribution_PDF.svg\|325px\|Plot of the Wigner semicircle PDF]]<br /><small></small>\|\|
			cdf_image =[[Image:WignerS_distribution_CDF.svg\|325px\|Plot of the Wigner semicircle CDF]]<br /><small></small>\|
			parameters =<math>R>0\!</math> [[radius]] ([[real number\|real]])\|
			support =<math>x \in [-R;+R]\!</math>\|
			pdf =<math>\frac2{\pi R^2}\,\sqrt{R^2-x^2}\!</math>\|
			cdf =<math>\frac12+\frac{x\sqrt{R^2-x^2}}{\pi R^2} + \frac{\arcsin\!\left(\frac{x}{R}\right)}{\pi}\!</math><br />for <math>-R\leq x \leq R</math>\|
			mean =<math>0\,</math>\|
			median =<math>0\,</math>\|
			mode =<math>0\,</math>\|
			variance =<math>\frac{R^2}{4}\!</math>\|
			skewness =<math>0\,</math>\|
			kurtosis =<math>-1\,</math>\|
			entropy =<math>\ln (\pi R) - \frac12 \,</math>\|
			mgf =<math>2\,\frac{I_1(R\,t)}{R\,t}</math>\|
			char =<math>2\,\frac{J_1(R\,t)}{R\,t}</math>\|
			}}
			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 [[normalizing constant\|normalized]] (so that it is really a semi-ellipse):

	~~Check out my web~~-~~site~~ ... ~~luke bryan concert tickets 2014~~ ([http://www.~~netpaw~~.~~org navigate here~~])		:<math>f(x)={2 \over \pi R^2}\sqrt{R^2-x^2\,}\, </math>

			for −''R'' ≤ ''x'' ≤ ''R'', and ''f''(''x'') = 0 if ''R'' < ''\|x\|''.

			This distribution arises as the limiting distribution of [[eigenvalues]] of many [[random matrices\|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'' = 2''RY'' – ''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 2''n''-th [[moment (mathematics)\|moment]] of this distribution is

			:<math>E(X^{2n})=\left({R \over 2}\right)^{2n} C_n\, </math>

			where ''X'' is any random variable with this distribution and ''C''<sub>''n''</sub> is the ''n''th [[Catalan number]]

			:<math>C_n={1 \over n+1}{2n \choose n},\, </math>

			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 <math>x=R\cos(\theta)</math> into the defining equation for the [[moment generating function]] it can be seen that:

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

			which can be solved (see Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_376.htm §9.6.18)]
			to yield:

			:<math>M(t)=2\,\frac{I_1(Rt)}{Rt}</math>

			where <math>I_1(z)</math> is the modified [[Bessel function]]. Similarly, the characteristic function is given by:

			:<math>\varphi(t)=2\,\frac{J_1(Rt)}{Rt}</math>

			where <math>J_1(z)</math> is the Bessel function. (See Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_360.htm §9.1.20)], noting that the corresponding integral involving <math>\sin(Rt\cos(\theta))</math> is zero.)

			In the limit of <math>R</math> approaching zero, the Wigner semicircle distribution becomes a [[Dirac delta function]].

			== 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 [[cumulant]]s is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all [[partition of a set\|partitions of a finite set]] in the theory of ordinary cumulants is replaced by the set of all [[noncrossing partition]]s of a finite set. Just 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.

			== See also ==

			* The W.s.d. is the limit of the [[Kesten–McKay measure\|Kesten–McKay distributions]], as the parameter ''d'' tends to infinity.
			* In [[number theory\|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. ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables.'' New York: Dover, 1972.

			==External links==
			*[[Eric W. Weisstein]] et al., [http://mathworld.wolfram.com/WignersSemicircleLaw.html Wigner's semicircle]

			{{ProbDistributions\|continuous-bounded}}

			[[Category:Continuous distributions]]
			[[Category:Probability distributions]]

en>Thewikicontributor: /* See also */

2012-08-12T08:05:56Z

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80.229.246.38: /* Characters */ "Also ... as well" is an unnecessary redundant redundancy which isn't needed ;)

en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9803)

en>Thewikicontributor: /* See also */