Rician distribution

Probability density function
Cumulative distribution function
Parameters	ν ≥ 0 — distance between the reference point and the center of the bivariate distribution, σ ≥ 0 — scale
Support	x ∈ [0, +∞)
PDF	${\displaystyle {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)}$
CDF	${\displaystyle 1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)}$ where Q1 is the Marcum Q-function
Mean	${\displaystyle \sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-\nu ^{2}/2\sigma ^{2})}$
Variance	${\displaystyle 2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)}$
Skewness	(complicated)
Ex. kurtosis	(complicated)

${\displaystyle 1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)}$

In probability theory, the Rice distribution, Rician distribution or Ricean distribution is the probability distribution of the magnitude of a circular bivariate normal random variable with potentially non-zero mean. It was named after Stephen O. Rice.

The probability density function is

where I₀(z) is the modified Bessel function of the first kind with order zero.

In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter ${\displaystyle K={\frac {\nu ^{2}}{2\sigma ^{2}}}}$, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter ${\displaystyle \Omega =\nu ^{2}+2\sigma ^{2}}$, defined as the total power received in all paths.