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Encyclopedia > Rice distribution
 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 Support pdf cdf Mean Median Mode Variance Skewness (complicated) Kurtosis (complicated) Entropy mgf Char. func.

where I0(z) is the modified Bessel function of the first kind. When v =0 the distribution reduces to a Rayleigh distribution. In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ...

Contents

The first few raw moments are: Moment refers to either of two related concepts in mathematics and physics: Moment (physics) Moment (mathematics) See also Moment (magazine), a Jewish general publication. ...

where Lν(x) are a Laguerre polynomial generalized to a real parameter. They may be expressed in terms of the confluent hypergeometric function (M) and the hypergeometric function (F) as: In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Generalization The orthogonality property stated above is equivalent to saying that if X is an exponentially distributed random... In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

Generally the moments are given by:

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

Limiting cases

For large values of the argument, the Laguerre polynomial becomes (See Abramowitz and Stegun §13.5.1)

It is seen that as v becomes large or σ becomes small the mean becomes v and the variance becomes v2.

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