# 莱斯分布

參數 Rice probability density functions for various v   with σ=1. Rice probability density functions for various v   with σ=0.25. 概率density函數 Rice cumulative density functions for various v   with σ=1. Rice cumulative density functions for various v   with σ=0.25. 累積分佈函數 $v\ge 0\,$ $\sigma\ge 0\,$ $x\in [0;\infty)$ $\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)$ $\sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)$ $2\sigma^2+v^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-v^2}{2\sigma^2}\right)$ (complicated) (complicated)

$f(x|v,\sigma)=\,$
$\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)$

## 极限情况

For large values of the argument, the Laguerre polynomial becomes (See Abramowitz and Stegun §13.5.1)

$\lim_{x\rightarrow -\infty}L_\nu(x)=\frac{|x|^\nu}{\Gamma(1+\nu)}$

It is seen that as $v$ becomes large or $\sigma$ becomes small the mean becomes $v$ and the variance becomes $\sigma^2$