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Rice

Probability density function Rice probability density functions for various v with σ=1. Rice probability density functions for various v with σ=0.25.
Cumulative distribution function Rice cumulative density functions for various v with σ=1. 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)
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.

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.

Properties

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, L_ν(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 = σ^1/2.

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

Related distributions

If $R \sim \mathrm{Rice}\left(1,v\right)$ then R² has a noncentral chi-square distribution with two degrees of freedom and noncentrality parameter v².

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.

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.

Proakis, J., Digital Communications, McGraw-Hill, 2000.

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