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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wigner semicircle distribution ： ウィキペディア英語版 Wigner semicircle distribution + \frac\right)}\! for $-R\backslash leq\; x\; \backslash leq\; R$| mean =$0\backslash ,$| median =$0\backslash ,$| mode =$0\backslash ,$| variance =$\backslash frac\backslash !$| skewness =$0\backslash ,$| kurtosis =$-1\backslash ,$| entropy =$\backslash ln\; (\backslash pi\; R)\; -\; \backslash frac12\; \backslash ,$| mgf =$2\backslash ,\backslash frac$| char =$2\backslash ,\backslash frac$| }} The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval (''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): :$f(x)=\backslash sqrt\backslash ,$ 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'' = 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 of this distribution is :$E(X^)=\backslash left(\backslash right)^\; C\_n\backslash ,$ where ''X'' is any random variable with this distribution and ''C''''n'' is the ''n''th Catalan number :$C\_n=,\backslash ,$ 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 $x=R\backslash cos(\backslash theta)$ into the defining equation for the moment generating function it can be seen that: :$M(t)=\backslash frac\backslash int\_0^\backslash pi\; e^\backslash sin^2(\backslash theta)\backslash ,d\backslash theta$ which can be solved (see Abramowitz and Stegun (§9.6.18) ) to yield: :$M(t)=2\backslash ,\backslash frac$ where $I\_1(z)$ is the modified Bessel function. Similarly, the characteristic function is given by: :$\backslash varphi(t)=2\backslash ,\backslash frac$ where $J\_1(z)$ is the Bessel function. (See Abramowitz and Stegun (§9.1.20) ), noting that the corresponding integral involving $\backslash sin(Rt\backslash cos(\backslash theta))$ is zero.) In the limit of $R$ approaching zero, the Wigner semicircle distribution becomes a Dirac delta function. Differential equation $$ \left\\right\} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wigner semicircle distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.