# Rice distribution

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
 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. $v\ge 0\,$$\sigma\ge 0\,$ $x\in [0;\infty)$ $\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)$ $1-Q_1\left(\frac{v}{\sigma },\frac{x}{\sigma }\right)$ Where $Q_1$ is the Marcum Q-Function $\sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)$ $2\sigma^2+v^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-v^2}{2\sigma^2}\right)$ (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:

$f(x|v,\sigma)=\,$
$\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)$

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:

$\mu_1= \sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)$
$\mu_2= 2\sigma^2+v^2\,$
$\mu_3= 3\sigma^3\sqrt{\pi/2}\,\,L_{3/2}(-v^2/2\sigma^2)$
$\mu_4= 8\sigma^4+8\sigma^2v^2+v^4\,$
$\mu_5=15\sigma^5\sqrt{\pi/2}\,\,L_{5/2}(-v^2/2\sigma^2)$
$\mu_6=48\sigma^6+72\sigma^4v^2+18\sigma^2v^4+v^6\,$
$L_\nu(x)=L_\nu^0(x)=M(-\nu,1,x)=\,_1F_1(-\nu;1;x)$

where, Lν(x) denotes a Laguerre polynomial.

For the case ν = 1/2:

$L_{1/2}(x)=\,_1F_1\left( -\frac{1}{2};1;x\right)$
$=e^{x/2} \left[\left(1-x\right)I_0\left(\frac{-x}{2}\right) -xI_1\left(\frac{-x}{2}\right) \right]$

Generally the moments are given by

$\mu_k=s^k2^{k/2}\,\Gamma(1\!+\!k/2)\,L_{k/2}(-v^2/2\sigma^2), \,$

where s = σ1/2.

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

## Related distributions

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

## Limiting cases

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

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

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