|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.
|Probability density function (pdf)|
|Cumulative distribution function (cdf)||
Where is the Marcum Q-Function
|Moment-generating function (mgf)|
The probability density function is:
The first few raw moments are:
where, Lν(x) denotes a Laguerre polynomial.
For the case ν = 1/2:
Generally the moments are given by
where s = σ1/2.
When k is even, the moments become actual polynomials in σ and v.
- has a Rice distribution if where and are two independent normal distributions and is any real number.
- Another case where comes from the following steps:
- 1. Generate having a Poisson distribution with parameter (also mean, for a Poisson)
- 2. Generate having a Chi-squared distribution with degrees of freedom.
- 3. Set
- If then has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter .
For large values of the argument, the Laguerre polynomial becomes (see Abramowitz and Stegun §13.5.1)
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)
- The SOCR Resource provides interactive Rice distribution, Rice simulation, model-fitting and parameter estimation.
- 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.