Rayleigh Distribution edit the Wikipedia entry

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Rayleigh

Probability density function
Cumulative distribution function
Parameters
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. It usually arises when a two-dimensional vector (e.g. wind velocity) has its two orthogonal components normally and independently distributed. The absolute value (e.g. wind speed) will then have a Rayleigh distribution. The distribution may also arise in the case of random complex numbers whose real and imaginary components are normally and independently distributed. The absolute value of these numbers will then be Rayleigh-distributed.

The probability density function is

The characteristic function is given by:

where is the complex error function. The moment generating function is given by

where erf(z) is the error function. The raw moments are then given by

where Γ(z) is the Gamma function. The moments may be used to calculate:

Mean:

Variance:

Skewness:

Kurtosis:

Parameter estimation

Given N independent and identically distributed Rayleigh random variables with parameter σ, the maximum likelihood estimate of σ is

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 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.

See also

• Rayleigh fading
• Rice distribution