RayleighDistribution

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

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:

Show is an ExponentialDistribution:

Find the moment-generating function (MGF):

Find the mean and MGF in terms of the mean:

Find the amount of fading:

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:

RayleighDistribution is a special case of GammaDistribution:

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:

The radial CDF: