A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane 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, which is infrequent, 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.
Consider the two-dimensional vector which has components that are bivariate normally distributed, centered at zero, and independent.[clarification needed] Then and have density functions
Let be the length of . That is, Then has cumulative distribution function
Finally, the probability density function for is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is
Generalization to bivariate Student's t-distribution
Suppose is a random vector with components that follows a multivariate t-distribution. If the components both have mean zero, equal variance, and are independent, the bivariate Student's-t distribution takes the form:
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 in the above parametrization of the Rayleigh density.
The Weibull distribution with the shape parameterk = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter is related to the Weibull scale parameter according to
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 Rayleigh distribution was also employed in the field of nutrition for linking dietarynutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship.[9]
In the field of ballistics, the Rayleigh distribution is used for calculating the circular error probable—a measure of a gun's precision.
^"The Wave Theory of Light", Encyclopedic Britannica 1888; "The Problem of the Random Walk", Nature 1905 vol.72 p.318
^ abPapoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processes. ISBN 0073660116, ISBN 9780073660110 [page needed]
^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.
^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
^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
^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. CiteSeerX10.1.1.18.1228. doi:10.1002/(sici)1098-1098(1999)10:2<109::aid-ima2>3.0.co;2-r.
^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. PMID 25059432.
^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. PMC5697816. PMID 29161271.
^"Rayleigh Probability Distribution Applied to Random Wave Heights" (PDF). United States Naval Academy.