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| Rayleigh Distribution edit the Wikipedia entry |
ODP's article on rayleigh distribution h
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In probability theory and statistics, the Rayleigh distribution (pronounced: /'reɪlɪ/) is a continuous probability distribution. As an example of how it arises, the wind speed will have a Rayleigh distribution if the components of the two-dimensional wind velocity vector are uncorrelated and normally distributed with equal variance. The distribution is named after Lord Rayleigh.
The Rayleigh probability density function is
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Contents
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Properties
The raw moments are given by:
where Γ(z) is the Gamma function.
The mean and variance of a Rayleigh random variable may be expressed as:
and
The mode is σ and the maximum pdf is
The skewness is given by:
The excess kurtosis is given by:
The characteristic function is given by:
where is the complex error function. The moment generating function is given by
where is the error function.
Information entropy
The information entropy is given by
where γ is the Euler–Mascheroni constant.
Parameter estimation
Given N independent and identically distributed Rayleigh random variables with parameter σ, the maximum likelihood estimate of σ is
Generating Rayleigh-distributed random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
has a Rayleigh distribution with parameter σ. This follows from the form of the cumulative distribution function. Given that U is uniform, (1–U) has the same uniformity and the above may be simplified to
Note that if you are generating random numbers belonging to [0,1), exclude zero values to avoid the natural log of zero.
Related distributions
- is a Rayleigh distribution if , where and are two independent normal distributions. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
- If R˜Rayleigh(1), then R2 has a chi-square distribution with two degrees of freedom:
- If X has an exponential distribution , then .
- If , then has a gamma distribution with parameters N and 2σ2: .
- The Chi distribution is a generalization of the Rayleigh distribution.
- The Rice distribution is a generalization of the Rayleigh distribution.
- The Weibull distribution is a generalization of the Rayleigh distribution.
- The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
References
See also
- Rayleigh fading
- Rice distribution
- The SOCR Resource provides interactive interface to Rayleigh distribution.