# 維格納半圓分布

參數 概率density函數 累積分佈函數 ${\displaystyle R>0\!}$ radius (real) ${\displaystyle x\in [-R;+R]\!}$ ${\displaystyle {\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!}$ ${\displaystyle {\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!}$ for ${\displaystyle -R\leq x\leq R}$ ${\displaystyle 0\,}$ ${\displaystyle 0\,}$ ${\displaystyle 0\,}$ ${\displaystyle {\frac {R^{2}}{4}}\!}$ ${\displaystyle 0\,}$ ${\displaystyle -1\,}$ ${\displaystyle \ln(\pi R)-{\frac {1}{2}}\,}$ ${\displaystyle 2\,{\frac {I_{1}(R\,t)}{R\,t}}}$ ${\displaystyle 2\,{\frac {J_{1}(R\,t)}{R\,t}}}$

${\displaystyle f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,}$

for −RxR, and f(x) = 0 if R < |x|.

## 性質

${\displaystyle E(X^{2n})=\left({R \over 2}\right)^{2n}C_{n}\,}$

${\displaystyle C_{n}={1 \over n+1}{2n \choose n},\,}$

(因為對稱性的關係，所有奇數項之動差皆為0)

${\displaystyle M(t)={\frac {2}{\pi }}\int _{0}^{\pi }e^{Rt\cos(\theta )}\sin ^{2}(\theta )\,d\theta }$

${\displaystyle M(t)=2\,{\frac {I_{1}(Rt)}{Rt}}}$

${\displaystyle \varphi (t)=2\,{\frac {J_{1}(Rt)}{Rt}}}$

${\displaystyle \left\{\left(r^{2}-x^{2}\right)f'(x)+xf(x)=0,f(1)={\frac {2{\sqrt {r^{2}-1}}}{\pi r^{2}}}\right\}}$