莱斯分布

在概率论与数理统计领域，莱斯分布（Rice distribution或Rician distribution）是一种连续概率分布，以美国科学家斯蒂芬·莱斯（英语：Stephen O. Rice）的名字命名，其概率密度函数为：

Rice

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

${\displaystyle f(x|v,\sigma )=\,}$

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

其中${\displaystyle I_{0}(z)}$是修正的第一类零阶贝塞尔函数（Bessel function）。当${\displaystyle v=0}$时，莱斯分布退化为瑞利分布。