Probability density function
Cumulative distribution function
Parameters	${\displaystyle \nu \geq 0}$, distance between the reference point and the center of the bivariate distribution, $\sigma \geq 0$, scale
Support	${\displaystyle x\in [0,\infty )}$
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)
Ex. 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).

Characterization

The probability density function is

$f(x\mid \nu ,\sigma )={\frac {x}{\sigma ^{2))}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2))}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2))}\right),$

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.^[1]

The characteristic function of the Rice distribution is given as:^[2][3]

${\displaystyle {\begin{aligned}\chi _{X}(t\mid \nu ,\sigma )=\exp \left(-{\frac {\nu ^{2)){2\sigma ^{2))}\right)&\left[\Psi _{2}\left(1;1,{\frac {1}{2));{\frac {\nu ^{2)){2\sigma ^{2))},-{\frac {1}{2))\sigma ^{2}t^{2}\right)\right.\\[8pt]&\left.{}+i{\sqrt {2))\sigma t\Psi _{2}\left({\frac {3}{2));1,{\frac {3}{2));{\frac {\nu ^{2)){2\sigma ^{2))},-{\frac {1}{2))\sigma ^{2}t^{2}\right)\right],\end{aligned))}$

where $\Psi _{2}\left(\alpha ;\gamma ,\gamma ';x,y\right)$ is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of $x$ and $y$. It is given by:^[4][5]

$\Psi _{2}\left(\alpha ;\gamma ,\gamma ';x,y\right)=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{\frac {(\alpha )_{m+n)){(\gamma )_{m}(\gamma ')_{n))}{\frac {x^{m}y^{n)){m!n!)),$

where

$(x)_{n}=x(x+1)\cdots (x+n-1)={\frac {\Gamma (x+n)}{\Gamma (x)))$

is the rising factorial.

Properties

Moments

The first few raw moments are:

${\displaystyle {\begin{aligned}\mu _{1}^{'}&=\sigma {\sqrt {\pi /2))\,\,L_{1/2}(-\nu ^{2}/2\sigma ^{2})\\\mu _{2}^{'}&=2\sigma ^{2}+\nu ^{2}\,\\\mu _{3}^{'}&=3\sigma ^{3}{\sqrt {\pi /2))\,\,L_{3/2}(-\nu ^{2}/2\sigma ^{2})\\\mu _{4}^{'}&=8\sigma ^{4}+8\sigma ^{2}\nu ^{2}+\nu ^{4}\,\\\mu _{5}^{'}&=15\sigma ^{5}{\sqrt {\pi /2))\,\,L_{5/2}(-\nu ^{2}/2\sigma ^{2})\\\mu _{6}^{'}&=48\sigma ^{6}+72\sigma ^{4}\nu ^{2}+18\sigma ^{2}\nu ^{4}+\nu ^{6}\end{aligned))}$

and, in general, the raw moments are given by

$\mu _{k}^{'}=\sigma ^{k}2^{k/2}\,\Gamma (1\!+\!k/2)\,L_{k/2}(-\nu ^{2}/2\sigma ^{2}).\,$

Here L_q(x) denotes a Laguerre polynomial:

$L_{q}(x)=L_{q}^{(0)}(x)=M(-q,1,x)=\,_{1}F_{1}(-q;1;x)$

where $M(a,b,z)=_{1}F_{1}(a;b;z)$ is the confluent hypergeometric function of the first kind. When k is even, the raw moments become simple polynomials in σ and ν, as in the examples above.

For the case q = 1/2:

${\displaystyle {\begin{aligned}L_{1/2}(x)&=\,_{1}F_{1}\left(-{\frac {1}{2));1;x\right)\\&=e^{x/2}\left[\left(1-x\right)I_{0}\left(-{\frac {x}{2))\right)-xI_{1}\left(-{\frac {x}{2))\right)\right].\end{aligned))}$

The second central moment, the variance, is

$\mu _{2}=2\sigma ^{2}+\nu ^{2}-(\pi \sigma ^{2}/2)\,L_{1/2}^{2}(-\nu ^{2}/2\sigma ^{2}).$

Note that $L_{1/2}^{2}(\cdot )$ indicates the square of the Laguerre polynomial $L_{1/2}(\cdot )$, not the generalized Laguerre polynomial $L_{1/2}^{(2)}(\cdot ).$

Related distributions

• ${\displaystyle R\sim \mathrm {Rice} \left(|\nu |,\sigma \right)}$ if $R={\sqrt {X^{2}+Y^{2))}$ where $X\sim N\left(\nu \cos \theta ,\sigma ^{2}\right)$ and $Y\sim N\left(\nu \sin \theta ,\sigma ^{2}\right)$ are statistically independent normal random variables and $\theta$ is any real number.
• Another case where $R\sim {\mathrm {Rice} }\left(\nu ,\sigma \right)$ comes from the following steps:
  1. Generate $P$ having a Poisson distribution with parameter (also mean, for a Poisson) $\lambda ={\frac {\nu ^{2)){2\sigma ^{2))}.$
  2. Generate $X$ having a chi-squared distribution with 2P + 2 degrees of freedom.
  3. Set $R=\sigma {\sqrt {X)).$
• If ${\displaystyle R\sim \operatorname {Rice} (\nu ,1)}$ then $R^{2}$ has a noncentral chi-squared distribution with two degrees of freedom and noncentrality parameter $\nu ^{2}$.
• If ${\displaystyle R\sim \operatorname {Rice} (\nu ,1)}$ then $R$ has a noncentral chi distribution with two degrees of freedom and noncentrality parameter $\nu$.
• If ${\displaystyle R\sim \operatorname {Rice} (0,\sigma )}$ then ${\displaystyle R\sim \operatorname {Rayleigh} (\sigma )}$, i.e., for the special case of the Rice distribution given by $\nu =0$, the distribution becomes the Rayleigh distribution, for which the variance is $\mu _{2}={\frac {4-\pi }{2))\sigma ^{2}$.
• If ${\displaystyle R\sim \operatorname {Rice} (0,\sigma )}$ then $R^{2}$ has an exponential distribution.^[6]
• If ${\displaystyle R\sim \operatorname {Rice} \left(\nu ,\sigma \right)}$ then $1/R$ has an Inverse Rician distribution.^[7]
• The folded normal distribution is the univariate special case of the Rice distribution.

Limiting cases

For large values of the argument, the Laguerre polynomial becomes^[8]

${\displaystyle \lim _{x\to -\infty }L_{\nu }(x)={\frac {|x|^{\nu )){\Gamma (1+\nu ))).}$

It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ².

The transition to a Gaussian approximation proceeds as follows. From Bessel function theory we have

${\displaystyle I_{\alpha }(z)\to {\frac {e^{z)){\sqrt {2\pi z))}\left(1-{\frac {4\alpha ^{2}-1}{8z))+\cdots \right){\text{ as ))z\rightarrow \infty }$

so, in the large ${\displaystyle x\nu /\sigma ^{2))$ region, an asymptotic expansion of the Rician distribution:

${\displaystyle {\begin{aligned}f(x,\nu ,\sigma )={}&{\frac {x}{\sigma ^{2))}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2))}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2))}\right)\\{\text{ is ))\\&{\frac {x}{\sigma ^{2))}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2))}\right){\sqrt {\frac {\sigma ^{2)){2\pi x\nu ))}\exp \left({\frac {2x\nu }{2\sigma ^{2))}\right)\left(1+{\frac {\sigma ^{2)){8x\nu ))+\cdots \right)\\\rightarrow {}&{\frac {1}{\sigma {\sqrt {2\pi ))))\exp \left(-{\frac {(x-\nu )^{2)){2\sigma ^{2))}\right){\sqrt {\frac {x}{\nu ))},\;\;\;{\text{ as )){\frac {x\nu }{\sigma ^{2))}\rightarrow \infty \end{aligned))}$

Moreover, when the density is concentrated around ${\textstyle \nu }$ and ${\textstyle |x-\nu |\ll \sigma }$ because of the Gaussian exponent, we can also write ${\textstyle {\sqrt ((x}/{\nu ))}\approx 1}$ and finally get the Normal approximation

${\displaystyle f(x,\nu ,\sigma )\approx {\frac {1}{\sigma {\sqrt {2\pi ))))\exp \left(-{\frac {(x-\nu )^{2)){2\sigma ^{2))}\right),\;\;\;{\frac {\nu }{\sigma ))\gg 1}$

The approximation becomes usable for ${\displaystyle {\frac {\nu }{\sigma ))>3}$

Parameter estimation (the Koay inversion technique)

There are three different methods for estimating the parameters of the Rice distribution, (1) method of moments,^{[9][10][11][12]} (2) method of maximum likelihood,^{[9][10][11][13]} and (3) method of least squares.^{[citation needed]} In the first two methods the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data. This can be done using the method of moments, e.g., the sample mean and the sample standard deviation. The sample mean is an estimate of μ₁^' and the sample standard deviation is an estimate of μ₂^1/2.

The following is an efficient method, known as the "Koay inversion technique".^[14] for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously . This inversion technique is also known as the fixed point formula of SNR. Earlier works^[9][15] on the method of moments usually use a root-finding method to solve the problem, which is not efficient.

First, the ratio of the sample mean to the sample standard deviation is defined as r, i.e., $r=\mu _{1}^{'}/\mu _{2}^{1/2}$. The fixed point formula of SNR is expressed as

$g(\theta )={\sqrt {\xi {(\theta )}\left[1+r^{2}\right]-2)),$

where $\theta$ is the ratio of the parameters, i.e., ${\displaystyle \theta ={\nu }/{\sigma ))$, and $\xi {\left(\theta \right)}$ is given by:

$\xi {\left(\theta \right)}=2+\theta ^{2}-{\frac {\pi }{8))\exp {(-\theta ^{2}/2)}\left[(2+\theta ^{2})I_{0}(\theta ^{2}/4)+\theta ^{2}I_{1}(\theta ^{2}/4)\right]^{2},$

where $I_{0}$ and $I_{1}$ are modified Bessel functions of the first kind.

Note that $\xi {\left(\theta \right)}$ is a scaling factor of $\sigma$ and is related to $\mu _{2}$ by:

${\displaystyle \mu _{2}=\xi {\left(\theta \right)}\sigma ^{2}.}$

To find the fixed point, $\theta ^{*}$, of $g$, an initial solution is selected, ${\theta }_{0}$, that is greater than the lower bound, which is ${\displaystyle {\theta }_{\text{lower bound))=0}$ and occurs when ${\textstyle r={\sqrt {\pi /(4-\pi )))}$^[14] (Notice that this is the $r=\mu _{1}^{'}/\mu _{2}^{1/2}$ of a Rayleigh distribution). This provides a starting point for the iteration, which uses functional composition,^{[clarification needed]} and this continues until $\left|g^{i}\left(\theta _{0}\right)-\theta _{i-1}\right|$ is less than some small positive value. Here, $g^{i}$ denotes the composition of the same function, $g$, $i$ times. In practice, we associate the final $\theta _{n}$ for some integer $n$ as the fixed point, $\theta ^{*}$, i.e., $\theta ^{*}=g\left(\theta ^{*}\right)$.

Once the fixed point is found, the estimates $\nu$ and $\sigma$ are found through the scaling function, $\xi {\left(\theta \right)}$, as follows:

$\sigma ={\frac {\mu _{2}^{1/2)){\sqrt {\xi \left(\theta ^{*}\right)))},$

and

$\nu ={\sqrt {\left(\mu _{1}^{'~2}+\left(\xi \left(\theta ^{*}\right)-2\right)\sigma ^{2}\right))).$

To speed up the iteration even more, one can use the Newton's method of root-finding.^[14] This particular approach is highly efficient.

Applications

• The Euclidean norm of a bivariate circularly-symmetric normally distributed random vector.
• Rician fading (for multipath interference))
• Effect of sighting error on target shooting.^[16]
• Analysis of diversity receivers in radio communications.^[17][18]
• Distribution of eccentricities for models of the inner Solar System after long-term numerical integration.^[19]