Rice distribution

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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 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 Q1 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.

Contents

[edit] Characterization

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.

[edit] Properties

[edit] Moments

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, Lν(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 = σ1/2.

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

[edit] Related distributions

  • 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.
  • Another case where R \sim \mathrm{Rice}\left(\sigma,v\right) comes from the following steps:
1. Generate P having a Poisson distribution with parameter (also mean, for a Poisson) \lambda = \frac{v^2}{2\sigma^2}.
2. Generate X having a Chi-squared distribution with 2P + 2 degrees of freedom.
3. Set R = \sigma\sqrt{X}.

[edit] Limiting cases

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.

[edit] See also

[edit] External links

[edit] References

  • 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.
v • d • e
Probability distributions
 
Continuous univariate supported on a semi-infinite interval