Wigner semicircle distribution について

|
}}
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)=\sqrt\,

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^)=\left(\right)^ C_n\,

where ''X'' is any random variable with this distribution and ''C''_''n'' is the ''n''th Catalan number
:

C_n=,\,

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\cos(\theta)

into the defining equation for the moment generating function it can be seen that:
:

M(t)=\frac\int_0^\pi e^\sin^2(\theta)\,d\theta

which can be solved (see Abramowitz and Stegun (§9.6.18) )
to yield:
:

M(t)=2\,\frac

where

I_1(z)

is the modified Bessel function. Similarly, the characteristic function is given by:
:

\varphi(t)=2\,\frac

where

J_1(z)

is the Bessel function. (See Abramowitz and Stegun (§9.1.20) ), noting that the corresponding integral involving

\sin(Rt\cos(\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