In probability theory, the Rice distribution or Rician distribution (or, less commonly, Ricean distribution) is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral).

The probability density function is where I0(z) is the modified Bessel function of the first kind with order zero.

In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter

, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter

, defined as the total power received in all paths.

[1] The characteristic function of the Rice distribution is given as:[2][3] where

is one of Horn's confluent hypergeometric functions with two variables and convergent for all finite values of

The first few raw moments are: and, in general, the raw moments are given by Here Lq(x) denotes a Laguerre polynomial: where

is the confluent hypergeometric function of the first kind.

When k is even, the raw moments become simple polynomials in σ and ν, as in the examples above.

For the case q = 1/2: The second central moment, the variance, is Note that

indicates the square of the Laguerre polynomial

, not the generalized Laguerre polynomial

For large values of the argument, the Laguerre polynomial becomes[8] It is seen that as ν becomes large or σ becomes small the mean becomes ν and the variance becomes σ2.

The transition to a Gaussian approximation proceeds as follows.

From Bessel function theory we have so, in the large

region, an asymptotic expansion of the Rician distribution: Moreover, when the density is concentrated around

because of the Gaussian exponent, we can also write

[citation needed] In the first two methods the interest is in estimating the parameters of the distribution, ν and σ, from a sample of data.

The following is an efficient method, known as the "Koay inversion technique".

[14] for solving the estimating equations, based on the sample mean and the sample standard deviation, simultaneously .

This inversion technique is also known as the fixed point formula of SNR.

Earlier works[9][15] on the method of moments usually use a root-finding method to solve the problem, which is not efficient.

The fixed point formula of SNR is expressed as where

are modified Bessel functions of the first kind.

, an initial solution is selected,

This provides a starting point for the iteration, which uses functional composition,[clarification needed] and this continues until

In practice, we associate the final

Once the fixed point is found, the estimates

, as follows: and To speed up the iteration even more, one can use the Newton's method of root-finding.