# 瑞利分布

參數 概率密函數 累積分佈函數 ${\displaystyle \sigma >0\,}$ ${\displaystyle x\in [0;\infty )}$ ${\displaystyle {\frac {x\exp \left({\frac {-x^{2}}{2\sigma ^{2}}}\right)}{\sigma ^{2}}}}$ ${\displaystyle 1-\exp \left({\frac {-x^{2}}{2\sigma ^{2}}}\right)}$ ${\displaystyle \sigma {\sqrt {\frac {\pi }{2}}}}$ ${\displaystyle \sigma {\sqrt {\ln(4)}}\,}$ ${\displaystyle \sigma \,}$ ${\displaystyle {\frac {4-\pi }{2}}\sigma ^{2}}$ ${\displaystyle {\frac {2{\sqrt {\pi }}(\pi -3)}{(4-\pi )^{3/2}}}}$ ${\displaystyle -{\frac {6\pi ^{2}-24\pi +16}{(4-\pi )^{2}}}}$ ${\displaystyle 1+\ln \left({\frac {\sigma }{\sqrt {2}}}\right)+{\frac {\gamma }{2}}}$ ${\displaystyle 1+\sigma t\,e^{\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\left({\textrm {erf}}\left({\frac {\sigma t}{\sqrt {2}}}\right)\!+\!1\right)}$ ${\displaystyle 1\!-\!\sigma te^{-\sigma ^{2}t^{2}/2}{\sqrt {\frac {\pi }{2}}}\!\left({\textrm {erfi}}\!\left({\frac {\sigma t}{\sqrt {2}}}\right)\!-\!i\right)}$

${\displaystyle f(x;\sigma )={\frac {x}{\sigma ^{2}}}e^{-x^{2}/2\sigma ^{2}},\quad x\geq 0,}$

## 參考文獻

1. ^ Athanasios Papoulis, S Pillai, "Probability, Random Variables and Stochastic Processes", 2001, ISBN 0073660116 / 9780073660110