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

# RayleighDistribution

 RayleighDistribution[] represents the Rayleigh distribution with scale parameter .
• The probability density for value in a Rayleigh distribution is proportional to for , and is zero for . »
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (6)
Generate a set of random numbers that are Rayleigh distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness and kurtosis are constant:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (7)
Consider vectors with standard normal components:
The angle will follow a uniform distribution:
The norm will follow a Rayleigh distribution:
A product has time to failure that is Rayleigh distributed with parameter . What is the product reliability at 4000, 4500, and 5000 hours? Reliability is also known as survival probability:
The lifetime of a device has a Rayleigh distribution. Find reliability of the device:
The failure rate increasing in time:
Find reliability of two such devices in series:
Find reliability of two such devices in parallel:
Compare reliability of both systems for and :
A vector has two components, which are normally distributed. Find the distribution of the length of the vector:
Find the average length of the vector:
Simulate possible lengths for a sample of 30 vectors:
RayleighDistribution can be used to approximate wind speeds:
Find the estimated distribution:
Compare the PDF to the histogram of the wind data:
Find the probability of a day with wind speed greater than 30 km/h:
Find the mean wind speed:
Simulate daily average wind speeds for a month:
Let be the mean of the highest one-third of the waves at a given site. The height of the waves at this site can be modeled by the RayleighDistribution:
Find the probability that a wave is higher than :
Find the average height of the waves at this site:
Assuming that meters, simulate the wave heights:
In the theory of fading channels, RayleighDistribution is used to model fading amplitude in the situation when no direct line of sight is present. 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 and MGF in terms of the mean:
Parameter influence on the CDF for each :
Rayleigh distribution is closed under scaling by a positive factor:
Relationships to other distributions:
RayleighDistribution with is a special case of ChiDistribution:
Square of RayleighDistribution with is a special case of ChiSquareDistribution:
Rayleigh distribution can be obtained as a transformation of ExponentialDistribution:
Rayleigh distribution is a special case of RiceDistribution:
BeniniDistribution is a transformation of Rayleigh distribution:
The norm of two standard normally distributed variables follows Rayleigh distribution:
Parameter mixture of NormalDistribution with Rayleigh distribution is LaplaceDistribution:
Rayleigh distribution is related to BinormalDistribution:
Rayleigh distribution is a special case of WeibullDistribution:
SuzukiDistribution can be obtained from LogNormalDistribution and Rayleigh distribution:
KDistribution can be represented as a parameter mixture of RayleighDistribution and GammaDistribution:
RayleighDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
Generate vectors with normal components:
The resulting bivariate distribution has a RayleighDistribution in its radial direction: