Rice distribution

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Rice
Probability density function Rice probability density functions for various v with σ=1. Rice probability density functions for various v with σ=0.25.
Cumulative distribution function Rice cumulative density functions for various v with σ=1. Rice cumulative density functions for various v with σ=0.25.
Parameters	$v\ge 0\,$ $\sigma\ge 0\,$
Support	$x\in [0;\infty)$
Probability density function (pdf)	$\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)$
Cumulative distribution function (cdf)
Mean	$\sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)$
Median
Mode
Variance	$2\sigma^2+v^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-v^2}{2\sigma^2}\right)$
Skewness	(complicated)
Excess kurtosis	(complicated)
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the Rice distribution is a continuous probability distribution. 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 $I 0 (z)$ is the modified Bessel function of the first kind with order zero. When v = 0 the distribution reduces to a Rayleigh distribution.

1 Moments
2 Related distributions
3 Limiting cases
4 See also
5 External links
6 References

[edit] 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, $L ν (x)$ 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 $s = σ 1 / 2$ .

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

[edit] 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 $θ$ is any real number.

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 $v 2$ .

[edit] 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 $σ 2$

[edit] See also

Rayleigh distribution
Stephen O. Rice (1907-1986)

[edit] External links

Yongjun Xie and Yuguang Fang, "A General Statistical Channel Model for Mobile Satellite Systems" IEEE Transactions on Vehicular Technology, VOL. 49, NO. 3, MAY 2000. http://www.fang.ece.ufl.edu/mypaper/tvt00_xie.pdf
MATLAB code for Rice distribtion (PDF, mean and variance, and generating random samples)

[edit] References

Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. ISBN 0-486-61272-4

J. Sijbers, Signal and noise estimation from magnetic resonance images, PhD thesis, p. 9-11, May 1998

Rice, S. O., Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945) 46-156.

Proakis, J., Digital Communications, McGraw-Hill, 2000.

Gudbjartsson, H.; S. Patz (1995). "The Rician distribution of noisy MRI data". Magn Reson Med 34: 910-4.

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