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A continuous [[Probability distribution|probability distribution]] with density
 
A continuous [[Probability distribution|probability distribution]] with density
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777301.png" /></td> </tr></table>
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p ( x)  = \
 
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depending on a [[Scale parameter|scale parameter]] $  \sigma > 0 $.  
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depending on a [[Scale parameter|scale parameter]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777302.png" />. A Rayleigh distribution has positive asymmetry; its unique mode is at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777303.png" />. All moments of a Rayleigh distribution are finite, the mathematical expectation and variance being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777305.png" />, respectively. The distribution function of a Rayleigh distribution has the form
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A Rayleigh distribution has positive asymmetry; its unique mode is at the point $  x = \sigma $.  
 
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All moments of a Rayleigh distribution are finite, the mathematical expectation and variance being $  \sigma \sqrt {\pi /2 } $
 
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and $  2 \sigma  ^ {2} ( 1 - \pi / 4 ) $,  
 
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respectively. The distribution function of a Rayleigh distribution has the form
 
  
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f ( x)  = \
 
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A Rayleigh distribution is a special case of the distribution with density
 
A Rayleigh distribution is a special case of the distribution with density
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777307.png" /></td> </tr></table>
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\frac{2}{2  ^ {n/2} \sigma  ^ {n} \Gamma ( n/2 ) }
 
  
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x  ^ {n-} 1 e ^ {- x  ^ {2} / 2 \sigma  ^ {2} }
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when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777308.png" />; hence, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r0777309.png" /> the Rayleigh distribution coincides with the distribution of the square root of a random variable which has the [[Chi-squared distribution| "chi-squared"  distribution]] with two degrees of freedom. In other words, a Rayleigh distribution can be interpreted as the distribution of the length of a vector in a plane Cartesian coordinate system, the coordinates of which are independent and have the [[Normal distribution|normal distribution]] with parameters 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077730/r07773010.png" />. In the three-dimensional space the [[Maxwell distribution|Maxwell distribution]] plays a role analogous to the Rayleigh distribution.
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$$
 
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when $  n = 2 $;  
 
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hence, when $  \sigma = 1 $
 
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the Rayleigh distribution coincides with the distribution of the square root of a random variable which has the [[Chi-squared distribution| "chi-squared"  distribution]] with two degrees of freedom. In other words, a Rayleigh distribution can be interpreted as the distribution of the length of a vector in a plane Cartesian coordinate system, the coordinates of which are independent and have the [[Normal distribution|normal distribution]] with parameters 0 and $  \sigma  ^ {2} $.  
 
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In the three-dimensional space the [[Maxwell distribution|Maxwell distribution]] plays a role analogous to the Rayleigh distribution.
 
  
 
A Rayleigh distribution is mainly applied in target theory and statistical communication theory. It was first considered by Lord Rayleigh in 1880 as the distribution of the amplitude resulting from the addition of harmonic oscillations.
 
A Rayleigh distribution is mainly applied in target theory and statistical communication theory. It was first considered by Lord Rayleigh in 1880 as the distribution of the amplitude resulting from the addition of harmonic oscillations.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. [Lord Rayleigh] Strutt,  "Wave theory of light" , Moscow-Leningrad  (1940)  (In Russian; translated from English)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W. [Lord Rayleigh] Strutt,  "Wave theory of light" , Moscow-Leningrad  (1940)  (In Russian; translated from English)</TD></TR></table>
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====Comments====
 
====Comments====
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Papoulis,  "Probability, random variables and stochastic processes" , McGraw-Hill  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Papoulis,  "Probability, random variables and stochastic processes" , McGraw-Hill  (1965)</TD></TR></table>

Revision as of 14:53, 7 June 2020

A continuous probability distribution with density

depending on a scale parameter . A Rayleigh distribution has positive asymmetry; its unique mode is at the point . All moments of a Rayleigh distribution are finite, the mathematical expectation and variance being and , respectively. The distribution function of a Rayleigh distribution has the form

A Rayleigh distribution is a special case of the distribution with density

when ; hence, when the Rayleigh distribution coincides with the distribution of the square root of a random variable which has the "chi-squared" distribution with two degrees of freedom. In other words, a Rayleigh distribution can be interpreted as the distribution of the length of a vector in a plane Cartesian coordinate system, the coordinates of which are independent and have the normal distribution with parameters 0 and . In the three-dimensional space the Maxwell distribution plays a role analogous to the Rayleigh distribution.

A Rayleigh distribution is mainly applied in target theory and statistical communication theory. It was first considered by Lord Rayleigh in 1880 as the distribution of the amplitude resulting from the addition of harmonic oscillations.

References

[1] J.W. [Lord Rayleigh] Strutt, "Wave theory of light" , Moscow-Leningrad (1940) (In Russian; translated from English)


Comments

References

[a1] A. Papoulis, "Probability, random variables and stochastic processes" , McGraw-Hill (1965)
How to Cite This Entry:
Rayleigh distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rayleigh_distribution&oldid=49391
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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