This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# RiceDistribution

 RiceDistribution represents a Rice distribution with shape parameters and . RiceDistributionrepresents a Norton-Rice distribution with parameters m, , and .
• The probability density for value in a Rice distribution is proportional to for , and is zero for .
• The probability density for value in a Norton-Rice distribution is proportional to for , and is zero for .
• RiceDistribution allows to be any non-negative real number and m, to be any positive real numbers.
Probability density function of Rice distribution:
Cumulative distribution function of Rice distribution:
Mean and variance of Rice distribution:
Probability density function of Norton-Rice distribution:
Cumulative distribution function of Norton-Rice distribution:
Mean and variance of Norton-Rice distribution:
Probability density function of Rice distribution:
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Cumulative distribution function of Rice distribution:
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Mean and variance of Rice distribution:
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Probability density function of Norton-Rice distribution:
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Cumulative distribution function of Norton-Rice distribution:
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Mean and variance of Norton-Rice distribution:
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 Scope   (6)
Generate a set of pseudorandom numbers that are Rice distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Skewness:
Kurtosis:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function of Rice distribution:
Hazard function of Norton-Rice distribution:
Quantile function of Rice distribution:
Quantile function of Norton-Rice distribution:
 Applications   (3)
A vector has two components, which are normally distributed with the same nonzero mean and the same variance . Assuming and , find the distribution of the length of the vector:
Plot the probability density function:
Find the average length of the vector:
Find the probability that the length is at least 4:
Simulate possible lengths for a sample of 30 vectors:
In the theory of fading channels, RiceDistribution is used to model fading amplitude in the situation when the signal is composed of one strong direct line of sight and many weaker random components. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise:
Find the moment-generating function (MGF):
Find the mean:
Express the MGF in terms of the mean:
Find the amount of fading:
Limiting value:
Consider vectors with normal components with mean 3 and standard deviation 0.5:
The norm will follow a Rice distribution:
Parameter influence on the CDF for each :
Rice distribution is closed under scaling by a positive factor:
Relationships to other distributions:
Norton-Rice distribution simplifies to Rice distribution for :
Rice distribution is the distribution of the norm of two variables with NormalDistribution:
Rice distribution is related to BinormalDistribution:
Rice-Norton distribution is related to MultinormalDistribution:
Rice distribution is a special case of BeckmannDistribution:
RayleighDistribution is a special case of Rice distribution:
NoncentralChiSquareDistribution can be obtained from Rice distribution:
A limit of a Norton-Rice distribution is a NakagamiDistribution:
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