Rice distribution

Probability distribution
name =Rice
type =density
pdf_

Rice probability density functions for various "ν" with σ = 0.25.
cdf_

Rice cumulative distribution functions for various "ν" with σ = 0.25.
parameters =$uge\; 0,$
$sigmage\; 0,$
support =$xin\; [0;infty)$
pdf =$frac\{x\}\{sigma^2\}expleft(frac\{-(x^2+\; u^2)\}\{2sigma^2\}\; ight)I\_0left(frac\{x\; u\}\{sigma^2\}\; ight)$
cdf =$1-Q\_1left(frac\{\; u\}\{sigma\; \},frac\{x\}\{sigma\; \}\; ight)$Where $Q\_1$ is the Marcum Q-Function
mean =$sigma\; sqrt\{pi/2\},,L\_\{1/2\}(-\; u^2/2sigma^2)$
median =
mode =
variance =$2sigma^2+\; u^2-frac\{pisigma^2\}\{2\}L\_\{1/2\}^2left(frac\{-\; u^2\}\{2sigma^2\}\; ight)$
skewness =(complicated)
kurtosis =(complicated)
entropy =
mgf =
char =

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

Characterization

The probability density function is:

:$f(x|\; u,sigma)\; =\; frac\{x\}\{sigma^2\}expleft(frac\{-(x^2+\; u^2)\}\{2sigma^2\}\; ight)I\_0left(frac\{x\; u\}\{sigma^2\}\; ight)$

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.

Properties

Moments

The first few raw moments are:

:$mu\_1=\; sigma\; sqrt\{pi/2\},,L\_\{1/2\}(-\; u^2/2sigma^2)$:$mu\_2=\; 2sigma^2+\; u^2,$:$mu\_3=\; 3sigma^3sqrt\{pi/2\},,L\_\{3/2\}(-\; u^2/2sigma^2)$:$mu\_4=\; 8sigma^4+8sigma^2\; u^2+\; u^4,$:$mu\_5=15sigma^5sqrt\{pi/2\},,L\_\{5/2\}(-\; u^2/2sigma^2)$:$mu\_6=48sigma^6+72sigma^4\; u^2+18sigma^2\; u^4+\; u^6,$:$L\_\; u(x)=L\_\; u^0(x)=M(-\; u,1,x)=,\_1F\_1(-\; u;1;x)$

where, "L"_ν("x") denotes a Laguerre polynomial.

For the case ν = 1/2:

:$L\_\{1/2\}(x)=,\_1F\_1left(\; -frac\{1\}\{2\};1;x\; ight)$:$=e^\{x/2\}\; left\; [left(1-x\; ight)I\_0left(frac\{-x\}\{2\}\; ight)\; -xI\_1left(frac\{-x\}\{2\}\; ight)\; ight]\; .$

Generally the moments are given by

:$mu\_k=s^k2^\{k/2\},Gamma(1!+!k/2),L\_\{k/2\}(-\; u^2/2sigma^2),\; ,$

where "s" = σ^1/2.

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

Related distributions

*$R\; sim\; mathrm\{Rice\}left(sigma,\; u\; ight)$ has a Rice distribution if $R\; =\; sqrt\{X^2\; +\; Y^2\}$ where $X\; sim\; Nleft(\; ucos\; heta,sigma^2\; ight)$ and $Y\; sim\; Nleft(\; u\; sin\; heta,sigma^2\; ight)$ are two independent normal distributions and $heta$ is any real number.

*Another case where $R\; sim\; mathrm\{Rice\}left(\; u,sigma\; ight)$ comes from the following steps:

:1. Generate $P$ having a Poisson distribution with parameter (also mean, for a Poisson) $lambda\; =\; frac\{\; u^2\}\{2sigma^2\}.$

:2. Generate $X$ having a Chi-squared distribution with 2"P" + 2 degrees of freedom.

:3. Set $R\; =\; sigmasqrt\{X\}.$

*If $R\; sim\; mathrm\{Rice\}left(1,\; u\; ight)$ then $R^2$ has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter $u^2$.

Limiting cases

For large values of the argument, the Laguerre polynomial becomes (see Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_508.htm §13.5.1] )

:$lim\_\{x\; ightarrow\; -infty\}L\_\; u(x)=frac\{|x|^\; u\}\{Gamma(1+\; u)\}.$

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

See also

* Rayleigh distribution
* Stephen O. Rice (1907–1986)
* The SOCR Resource provides [http://socr.ucla.edu/htmls/SOCR_Distributions.html interactive Rice distribution] , [http://socr.ucla.edu/htmls/SOCR_Modeler.html Rice simulation, model-fitting and parameter estimation] .

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
* Rice, S. O., Mathematical Analysis of Random Noise. Bell System Technical Journal 24 (1945) 46–156.
* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WM3-4PK8B4Y-7&_user=1067359&_coverDate=11%2F20%2F2007&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000051243&_version=1&_urlVersion=0&_userid=1067359&md5=7ced136019d5f6faa50131ea0d21b3c9 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.

External links

* [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=14237&objectType=FILE MATLAB code for Rice distribtion] (PDF, mean and variance, and generating random samples)