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)	$1-Q_1\left(\frac{v}{\sigma },\frac{x}{\sigma }\right)$ Where $Q 1$ is the Marcum Q-Function
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, named after Stephen O. Rice, is a continuous probability distribution.

1 Characterization
2 Properties
- 2.1 Moments
3 Related distributions
4 Limiting cases
5 See also
6 External links
7 References

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.

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.

$R \sim \mathrm{Rice}\left(\sigma,v\right)$ 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 $v 2$ .

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.

Rayleigh distribution
Stephen O. Rice (1907-1986)

MATLAB code for Rice distribtion (PDF, mean and variance, and generating random samples)

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

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

I. Soltani Bozchalooi and Ming Liang, A smoothness index-guided approach to wavelet parameter selection in signal de-noising and fault detection, Journal of Sound and Vibration, Volume 308, Issues 1-2, 20 November 2007, Pages 246-267.

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

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Retrieved from "http://en.wikipedia.org/wiki/Rice_distribution"

Category: Continuous distributions

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