Rice distribution

In the 2D plane, pick a fixed point at distance ν from the origin. Generate a distribution of 2D points centered around that point, where the x and y coordinates are chosen independently from a Gaussian distribution with standard deviation σ (blue region). If R is the distance from these points to the origin, then R has a Rice distribution.

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)

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).