Rice distribution

Rice

Probability density function Rice probability density functions σ=1.0 Rice probability density functions for various v   with σ=1. Rice probability density functions σ=0.25 Rice probability density functions for various v   with σ=0.25.
Cumulative distribution function Rice cumulative density functions σ=1.0 Rice cumulative density functions for various v   with σ=1. Rice cumulative density functions σ=0.25 Rice cumulative density functions for various v   with σ=0.25.
Parameters	<math>v\ge 0\,</math> <math>\sigma\ge 0\,</math>
Support	<math>x\in [0;\infty)</math>
Probability density function (pdf)	<math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)} {2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)</math>
Cumulative distribution function (cdf)	<math>1-Q_1\left(\frac{v}{\sigma },\frac{x}{\sigma }\right)</math> Where <math>Q_1</math> is the Marcum Q-Function
Mean	<math>\sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)</math>
Median
Mode
Variance	<math>2\sigma^2+v^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-v^2}{2\sigma^2}\right)</math>
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.

Characterization

The probability density function is:

<math>f(x|v,\sigma)=\,</math>

<math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+v^2)}

{2\sigma^2}\right)I_0\left(\frac{xv}{\sigma^2}\right)</math>

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:

<math>\mu_1= \sigma \sqrt{\pi/2}\,\,L_{1/2}(-v^2/2\sigma^2)</math>

<math>\mu_2= 2\sigma^2+v^2\,</math>

<math>\mu_3= 3\sigma^3\sqrt{\pi/2}\,\,L_{3/2}(-v^2/2\sigma^2)</math>

<math>\mu_4= 8\sigma^4+8\sigma^2v^2+v^4\,</math>

<math>\mu_5=15\sigma^5\sqrt{\pi/2}\,\,L_{5/2}(-v^2/2\sigma^2)</math>

<math>\mu_6=48\sigma^6+72\sigma^4v^2+18\sigma^2v^4+v^6\,</math>

<math>L_\nu(x)=L_\nu^0(x)=M(-\nu,1,x)=\,_1F_1(-\nu;1;x)</math>

where, L_ν(x) denotes a Laguerre polynomial.

For the case ν = 1/2:

<math>L_{1/2}(x)=\,_1F_1\left( -\frac{1}{2};1;x\right)</math>

<math>=e^{x/2} \left[\left(1-x\right)I_0\left(\frac{-x}{2}\right) -xI_1\left(\frac{-x}{2}\right) \right]</math>

Generally the moments are given by

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

where s = σ^1/2.

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

Related distributions

• <math>R \sim \mathrm{Rice}\left(\sigma,v\right)</math> has a Rice distribution if <math>R = \sqrt{X^2 + Y^2}</math> where <math>X \sim N\left(v\cos\theta,\sigma^2\right)</math> and <math>Y \sim N\left(v \sin\theta,\sigma^2\right)</math> are two independent normal distributions and <math>\theta</math> is any real number.

• Another case where <math>R \sim \mathrm{Rice}\left(\sigma,v\right)</math> comes from the following steps:

1. Generate <math>P</math> having a Poisson distribution with parameter (also mean, for a Poisson) <math>\lambda = \frac{v^2}{2\sigma^2}.</math>

2. Generate <math>X</math> having a Chi-squared distribution with <math>2P + 2</math> degrees of freedom.

3. Set <math>R = \sigma\sqrt{X}.</math>

• If <math>R \sim \mathrm{Rice}\left(1,v\right)</math> then <math>R^2</math> has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter <math>v^2</math>.

Limiting cases

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

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

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

See also

• Rayleigh distribution
• Stephen O. Rice (1907-1986)
• The SOCR Resource provides interactive Rice distribution, Rice simulation, model-fitting and parameter estimation.

External links

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

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.

• 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.

it:Variabile casuale di Rice