Rayleigh distribution

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Probability density function
Plot of the Rayleigh PDF
Cumulative distribution function
Plot of the Rayleigh CDF
Probability density function (pdf)
Cumulative distribution function (cdf)
Excess kurtosis
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. It can arise when a two-dimensional vector (e.g. wind velocity) has elements that are normally distributed, are uncorrelated, and have equal variance. The vector’s magnitude (e.g. wind speed) will then have a Rayleigh distribution. The distribution can also arise in the case of random complex numbers whose real and imaginary components are i.i.d. Gaussian. In that case, the modulus of the complex number is Rayleigh-distributed. The distribution was so named after Lord Rayleigh.

The Rayleigh probability density function is



The raw moments are given by:

where is the Gamma function.

The mean and variance of a Rayleigh random variable may be expressed as:


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 is the Euler–Mascheroni constant.

Parameter estimation

Given N independent and identically distributed Rayleigh random variables with parameter , the maximum likelihood estimate of 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 has a chi-square distribution with two degrees of freedom:
  • If has an exponential distribution then .
  • If then has a gamma distribution with parameters and : .

See also

de:Rayleighverteilung fa:توزیع ریلی he:התפלגות ריילי it:Variabile casuale di Rayleigh ko:레일리 분포