# Rice distribution

Parameters Probability density functionRice probability density functions σ=1.0Rice probability density functions for various v   with σ=1.Rice probability density functions σ=0.25Rice probability density functions for various v   with σ=0.25. Cumulative distribution functionRice cumulative density functions σ=1.0Rice cumulative density functions for various v   with σ=1.Rice cumulative density functions σ=0.25Rice cumulative density functions for various v   with σ=0.25. ${\displaystyle v\geq 0\,}$${\displaystyle \sigma \geq 0\,}$ ${\displaystyle x\in [0;\infty )}$ ${\displaystyle {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+v^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {xv}{\sigma ^{2}}}\right)}$ ${\displaystyle 1-Q_{1}\left({\frac {v}{\sigma }},{\frac {x}{\sigma }}\right)}$ Where ${\displaystyle Q_{1}}$ is the Marcum Q-Function ${\displaystyle \sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-v^{2}/2\sigma ^{2})}$ ${\displaystyle 2\sigma ^{2}+v^{2}-{\frac {\pi \sigma ^{2}}{2}}L_{1/2}^{2}\left({\frac {-v^{2}}{2\sigma ^{2}}}\right)}$ (complicated) (complicated)

In probability theory and statistics, the Rice distribution, named after Stephen O. Rice, is a continuous probability distribution.

## Characterization

The probability density function is:

${\displaystyle f(x|v,\sigma )=\,}$
${\displaystyle {\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+v^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {xv}{\sigma ^{2}}}\right)}$

where I0(z) is the modified Bessel function of the first kind with order zero. When v = 0, the distribution reduces to a Rayleigh distribution.

## Properties

### Moments

The first few raw moments are:

${\displaystyle \mu _{1}=\sigma {\sqrt {\pi /2}}\,\,L_{1/2}(-v^{2}/2\sigma ^{2})}$
${\displaystyle \mu _{2}=2\sigma ^{2}+v^{2}\,}$
${\displaystyle \mu _{3}=3\sigma ^{3}{\sqrt {\pi /2}}\,\,L_{3/2}(-v^{2}/2\sigma ^{2})}$
${\displaystyle \mu _{4}=8\sigma ^{4}+8\sigma ^{2}v^{2}+v^{4}\,}$
${\displaystyle \mu _{5}=15\sigma ^{5}{\sqrt {\pi /2}}\,\,L_{5/2}(-v^{2}/2\sigma ^{2})}$
${\displaystyle \mu _{6}=48\sigma ^{6}+72\sigma ^{4}v^{2}+18\sigma ^{2}v^{4}+v^{6}\,}$
${\displaystyle L_{\nu }(x)=L_{\nu }^{0}(x)=M(-\nu ,1,x)=\,_{1}F_{1}(-\nu ;1;x)}$

where, Lν(x) denotes a Laguerre polynomial.

For the case ν = 1/2:

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

Generally the moments are given by

${\displaystyle \mu _{k}=s^{k}2^{k/2}\,\Gamma (1\!+\!k/2)\,L_{k/2}(-v^{2}/2\sigma ^{2}),\,}$

where s = σ1/2.

When k is even, the moments become actual polynomials in σ and v.

## Related distributions

• ${\displaystyle R\sim \mathrm {Rice} \left(\sigma ,v\right)}$ has a Rice distribution if ${\displaystyle R={\sqrt {X^{2}+Y^{2}}}}$ where ${\displaystyle X\sim N\left(v\cos \theta ,\sigma ^{2}\right)}$ and ${\displaystyle Y\sim N\left(v\sin \theta ,\sigma ^{2}\right)}$ are two independent normal distributions and ${\displaystyle \theta }$ is any real number.
• Another case where ${\displaystyle R\sim \mathrm {Rice} \left(\sigma ,v\right)}$ comes from the following steps:
1. Generate ${\displaystyle P}$ having a Poisson distribution with parameter (also mean, for a Poisson) ${\displaystyle \lambda ={\frac {v^{2}}{2\sigma ^{2}}}.}$
2. Generate ${\displaystyle X}$ having a Chi-squared distribution with ${\displaystyle 2P+2}$ degrees of freedom.
3. Set ${\displaystyle R=\sigma {\sqrt {X}}.}$
• If ${\displaystyle R\sim \mathrm {Rice} \left(1,v\right)}$ then ${\displaystyle R^{2}}$ has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter ${\displaystyle v^{2}}$.

## Limiting cases

For large values of the argument, the Laguerre polynomial becomes (see Abramowitz and Stegun §13.5.1)

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

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