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   Support   pdf   cdf   Mean   Median   Mode   Variance   Skewness  (complicated)  Kurtosis  (complicated)  Entropy   mgf   Char. func.   In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. The probability density function is: Image File history File links Download high resolution version (1300x975, 212 KB) Probability distribution function for the Rice distribution sigma=1. ...
Image File history File links Download high resolution version (1300x975, 156 KB) Probability distribution function for the Rice distribution sigma=0. ...
Image File history File links Download high resolution version (1300x975, 160 KB) Cumulative distribution function for the Rice distribution sigma=1. ...
Image File history File links Download high resolution version (1300x975, 138 KB) Cumulative distribution function for the Rice distribution sigma=0. ...
In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a realvalued random variable, X. For every real number x, the cdf is given by where the righthand side represents the probability that the variable X takes on a value less than or...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a realvalued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a realvalued random variable. ...
Entropy of a Bernoulli trial as a function of success probability. ...
In probability theory and statistics, the momentgenerating function of a random variable X is The momentgenerating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
Probability theory is the mathematical study of probability. ...
Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

where I_{0}(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. ...
Moments
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 L_{ν}(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/an1 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 v^{2}.
External links References 