# 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 ${\displaystyle v\geq 0\,}$ ${\displaystyle \sigma \geq 0\,}$ ${\displaystyle x\in [0;\infty )}$ ${\displaystyle {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+v^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {xv}{\sigma ^{2}}}\right)}$ ${\displaystyle \sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-v^{2}/2\sigma ^{2})}$ ${\displaystyle 2\sigma ^{2}+v^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-v^{2}}{2\sigma ^{2}}}\right)}$ (complicated) (complicated)

${\displaystyle f(x|v,\sigma )=\,}$
${\displaystyle {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+v^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {xv}{\sigma ^{2}}}\right)}$

## 极限情况

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

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

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