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Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for positive-valued random variables. It is a chi distribution in two degrees of freedom.

Rayleigh
Probability density function
Plot of the Rayleigh PDF
Cumulative distribution function
Plot of the Rayleigh CDF
Parametersscale:
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

The distribution is named after Lord Rayleigh (/ˈrli/).[1]

Contents

DefinitionEdit

The probability density function of the Rayleigh distribution is[2]

 

where   is the scale parameter of the distribution. The cumulative distribution function is[2]

 

for  

Relation to random vector lengthEdit

Consider the two-dimensional vector   which has components that are normally distributed, centered at zero, and independent. Then   and   have density functions

 

Let   be the length of  . That is,   Then   has cumulative distribution function

 

where   is the disk

 

Writing the double integral in polar coordinates, it becomes

 

Finally, the probability density function for   is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is

 

which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2. There are also generalizations when the components have unequal variance or correlations, or when the vector Y follows a bivariate Student t-distribution.[3]

PropertiesEdit

The raw moments are given by:

 

where   is the gamma function.

The mean of a Rayleigh random variable is thus :

 

The variance of a Rayleigh random variable is :

 

The mode is   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 imaginary error function. The moment generating function is given by

 

where   is the error function.

Differential entropyEdit

The differential entropy is given by[citation needed]

 

where   is the Euler–Mascheroni constant.

Parameter estimationEdit

Given a sample of N independent and identically distributed Rayleigh random variables   with parameter  ,

  is the maximum likelihood estimate and also is unbiased.
  is a biased estimator that can be corrected via the formula
 [4]

Confidence intervalsEdit

To find the (1 − α) confidence interval, first find the bounds   where:

   

then the scale parameter will fall within the bounds

   [5]

Generating random variatesEdit

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

 

has a Rayleigh distribution with parameter  . This is obtained by applying the inverse transform sampling-method.

Related distributionsEdit

  •   is Rayleigh distributed if  , where   and   are independent normal random variables.[6] (This gives motivation to the use of the symbol "sigma" in the above parametrization of the Rayleigh density.)
  • The chi distribution with v = 2 is equivalent to the Rayleigh Distribution with σ = 1. I.e., if  , then   has a chi-squared distribution with parameter  , degrees of freedom, equal to two (N = 2)
 
  • If  , then   has a gamma distribution with parameters   and  
 
  • The Rice distribution is a generalization of the Rayleigh distribution:  .
  • The Weibull distribution with the "shape parameter" k=2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter   is related to the Weibull scale parameter according to  
  • The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
  • If   has an exponential distribution  , then  

ApplicationsEdit

An application of the estimation of σ can be found in magnetic resonance imaging (MRI). As MRI images are recorded as complex images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.[7][8] The principle of Rayleigh distribution was also employed in the field of Nutrition for linking dietary nutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship.[9]

See alsoEdit

ReferencesEdit

  1. ^ "The Wave Theory of Light", Encyclopedic Britannica 1888; "The Problem of the Random Walk", Nature 1905 vol.72 p.318
  2. ^ a b Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110[page needed]
  3. ^ Röver, C. (2011). "Student-t based filter for robust signal detection". Physical Review D. 84 (12): 122004. arXiv:1109.0442. Bibcode:2011PhRvD..84l2004R. doi:10.1103/physrevd.84.122004.
  4. ^ Siddiqui, M. M. (1964) "Statistical inference for Rayleigh distributions", The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9, p. 1007
  5. ^ Siddiqui, M. M. (1961) "Some Problems Connected With Rayleigh Distributions", The Journal of Research of the National Bureau of Standards; Sec. D: Radio Propagation, Vol. 66D, No. 2, p. 169
  6. ^ Hogema, Jeroen (2005) "Shot group statistics"
  7. ^ Sijbers, J.; den Dekker, A. J.; Raman, E.; Van Dyck, D. (1999). "Parameter estimation from magnitude MR images". International Journal of Imaging Systems and Technology. 10 (2): 109–114. doi:10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r.
  8. ^ den Dekker, A. J.; Sijbers, J. (2014). "Data distributions in magnetic resonance images: a review". Physica Medica. 30 (7): 725–741. doi:10.1016/j.ejmp.2014.05.002.
  9. ^ Ahmadi, Hamed (2017-11-21). "A mathematical function for the description of nutrient-response curve". PLOS ONE. 12 (11): e0187292. Bibcode:2017PLoSO..1287292A. doi:10.1371/journal.pone.0187292. ISSN 1932-6203.