Rice distribution

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

$R \sim \mathrm{Rice}\left(\sigma,v\right)$ 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.

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

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.

	Probability distributions []
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • Infinite divisibility (probability) • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Rice distribution

Rice distribution

Characterization

Properties

Moments

Related distributions

Limiting cases

See also

External links

References

Rice distribution