represents the Rayleigh distribution with scale parameter σ.
- 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
- RayleighDistribution[σ] 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 RayleighDistribution are identical to the PDF of ChiDistribution and the CDF of ChiSquareDistribution, respectively. RayleighDistribution can be realized as a special case of RiceDistribution, GammaDistribution, and WeibullDistribution, as the PDF of RayleighDistribution[σ] 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.
Examplesopen allclose all
Basic Examples (4)
RayleighDistribution can be used to approximate wind speeds:
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:
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:
Properties & Relations (14)
Rayleigh distribution is a special case of RiceDistribution:
BeniniDistribution is a transformation of Rayleigh distribution:
Rayleigh distribution is related to BinormalDistribution:
Rayleigh distribution is a special case of WeibullDistribution:
Rayleigh distribution can be obtained as a transformation of ExponentialDistribution:
Possible Issues (2)
RayleighDistribution is not defined when σ is not a positive real number:
Neat Examples (2)
The resulting bivariate distribution has a RayleighDistribution in its radial direction: