Rice 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	${\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)$ ${\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)$
CDF	$1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)$ $1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)$ where Q₁ is the Marcum Q-function
Mean	$\sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-\nu ^{2}/2\sigma ^{2})$ $\sigma {\sqrt {\pi /2}}\,\,L_{{1/2}}(-\nu ^{2}/2\sigma ^{2})$
Variance	$2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)$ $2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{{1/2}}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)$
Skewness	(complicated)
Ex. kurtosis	(complicated)

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

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 $K={\frac {\nu ^{2}}{2\sigma ^{2}}}$ $K={\frac {\nu ^{2}}{2\sigma ^{2}}}$ , defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter $\Omega =\nu ^{2}+2\sigma ^{2}$ $\Omega =\nu ^{2}+2\sigma ^{2}$ , defined as the total power received in all paths.

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