 # 莱斯分布

参数 Rice probability density functions for various v   with σ=1. Rice probability density functions for various v   with σ=0.25.概率density函数 Rice cumulative density functions for various v   with σ=1. Rice cumulative density functions for various v   with σ=0.25.累积分布函数 $v\geq 0\,$ $\sigma \geq 0\,$ $x\in [0;\infty )$ ${\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+v^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {xv}{\sigma ^{2}}}\right)$ $\sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-v^{2}/2\sigma ^{2})$ $2\sigma ^{2}+v^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-v^{2}}{2\sigma ^{2}}}\right)$ (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 $\sigma ^{2}$ 