Rice distribution

	Probability density function
	Cumulative distribution function
Notation
Parameters	, distance between the reference point and the center of the bivariate distribution,; , scale
Support
PDF
CDF	where Q1 is the Marcum Q-function
Mean
Variance
Skewness	(complicated)
Excess kurtosis	(complicated)

In probability theory, the Rice distribution or Rician distribution (or, less commonly, Ricean distribution) is the probability distribution of the magnitude of a circularly symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice (1907–1986).

Probability density function
Cumulative distribution function
Notation	$\mathrm {Rice} (\nu ,\sigma )$
Parameters	$\nu \geq 0$ , distance between the reference point and the center of the bivariate distribution, $\sigma \geq 0$ , scale
Support	$x\in [0,\infty )$
PDF	${\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right)$
CDF	$1-Q_{1}\left({\frac {\nu }{\sigma }},{\frac {x}{\sigma }}\right)$ where Q₁ is the Marcum Q-function
Mean	$\sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-\nu ^{2}/2\sigma ^{2})$
Variance	$2\sigma ^{2}+\nu ^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-\nu ^{2}}{2\sigma ^{2}}}\right)$
Skewness	(complicated)
Excess kurtosis	(complicated)