Rice distribution

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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

Skewness (complicated)
Excess kurtosis (complicated)
Moment-generating function (mgf)
Characteristic function

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


The probability density function is:

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:

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.

Related distributions

  • 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 .

Limiting cases

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.

See also

External links


  • 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