# RayleighDistribution

represents the Rayleigh distribution with scale parameter σ.

# Details • The probability density for value in a Rayleigh distribution is proportional to for , and is zero for . »
• RayleighDistribution allows σ to be any positive real number.
• RayleighDistribution allows σ to be a quantity of any unit dimension. »
• RayleighDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

# Background & Context

• represents a continuous statistical distribution supported on the interval and parametrized by the positive real number σ (called a "scale parameter") that determines the overall behavior of its probability density function (PDF). In general, the PDF of a Rayleigh distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the value of σ. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• The Rayleigh distribution was first derived by its namesake, Lord Rayleigh, in the early 1880s as a tool for solving a particular problem in acoustics. In mathematics, the Rayleigh distribution is the probability distribution of the distance from the origin to a point in whenever the variables are all independent and identically distributed normal variates. Moreover, the Rayleigh distribution has been proven to model a large variety of phenomena, including two-dimensional random walks and manufacturing defects in electrovacuum devices, and is the distribution of the distance between an individual and its nearest neighbor in spatial configurations generated by Poisson processes.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Rayleigh distribution. Distributed[x,RayleighDistribution[σ]], written more concisely as xRayleighDistribution[σ], can be used to assert that a random variable x is distributed according to a Rayleigh distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for Rayleigh distributions may be given using PDF[RayleighDistribution[σ],x] and CDF[RayleighDistribution[σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Rayleigh distribution, EstimatedDistribution to estimate a Rayleigh parametric distribution from given data, and FindDistributionParameters to fit data to a Rayleigh distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Rayleigh distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Rayleigh distribution.
• TransformedDistribution can be used to represent a transformed Rayleigh distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Rayleigh distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Rayleigh distributions.
• RayleighDistribution is related to a number of other distributions. RayleighDistribution is related to ChiDistribution and ChiSquareDistribution in the sense that the PDF and CDF of are identical to the PDF of and the CDF of , respectively. RayleighDistribution can be realized as a special case of RiceDistribution, GammaDistribution, and WeibullDistribution, as the PDF of is equivalent to that of RiceDistribution[0,σ], GammaDistribution[1,σ , 2, 0], and WeibullDistribution[2,σ ], and moreover as a transformation (TransformedDistribution) of both ExponentialDistribution and BeniniDistribution. RayleighDistribution is also related to NormalDistribution, BinormalDistribution, LaplaceDistribution, SuzukiDistribution, LogNormalDistribution, and KDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

## Scope(7)

Generate a sample of random numbers from a Rayleigh distribution:

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:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the expected service time:

## Applications(7)

Consider vectors with standard normal components:

The angle will follow a uniform distribution:

The norm follows 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:

## Properties & Relations(14)

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 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:

Rayleigh distribution can be obtained as a transformation of ExponentialDistribution:

It is a special case of a Rayleigh distribution:

SuzukiDistribution can be obtained from LogNormalDistribution and Rayleigh distribution:

KDistribution can be represented as a parameter mixture of RayleighDistribution and GammaDistribution:

## Possible Issues(2)

RayleighDistribution is not defined when σ is not a positive real number: Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

## Neat Examples(2)

Generate vectors with normal components:

The resulting bivariate distribution has a RayleighDistribution in its radial direction: