Rice distribution

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.

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&nu;(x) denotes a Laguerre polynomial.

For the case &nu; = 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 = &sigma;1/2.

When k is even, the moments become actual polynomials in &sigma; 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 $$\theta$$ 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 $$R = \sigma\sqrt{X}.$$


 * 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 &sigma; becomes small the mean becomes v and the variance becomes &sigma;2.