Editing Rayleigh distribution (section)

==Relation to random vector length==

Consider the two-dimensional vector <math> Y = (U,V) </math> which has components that are [[bivariate normal distribution|bivariate normally distributed]], centered at zero, with equal variances <math>\sigma^2</math>, and independent. Then <math>U</math> and <math>V</math> have density functions

:<math>f_U(x; \sigma) = f_V(x;\sigma) = \frac{e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi\sigma^2}}.</math>

Let <math>X</math> be the length of <math>Y</math>. That is, <math>X = \sqrt{U^2 + V^2}.</math>  Then <math>X</math> has cumulative distribution function

:<math>F_X(x; \sigma) = \iint_{D_x} f_U(u;\sigma) f_V(v;\sigma) \,dA,</math>

where <math>D_x</math> is the disk

:<math>D_x = \left\{(u,v) : \sqrt{u^2 + v^2} \leq x\right\}.</math>

Writing the [[multiple integral|double integral]] in [[polar coordinate system|polar coordinates]], it becomes

:<math>F_X(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_0^{2\pi} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr\,d\theta = \frac 1 {\sigma^2} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr.
</math>

Finally, the probability density function for <math>X</math> is the derivative of its cumulative distribution function, which by the [[fundamental theorem of calculus]] is

:<math>f_X(x;\sigma) = \frac d {dx} F_X(x;\sigma) = \frac x {\sigma^2} e^{-x^2/(2\sigma^2)},</math>

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 ([[Hoyt distribution]]), or when the vector ''Y'' follows a [[multivariate t-distribution|bivariate Student ''t''-distribution]] (see also: [[Hotelling's T-squared distribution]]).<ref>{{cite journal|last=Röver|first=C.|title=Student-t based filter for robust signal detection|journal=Physical Review D|volume=84|issue=12|year=2011|pages=122004|doi=10.1103/physrevd.84.122004|arxiv=1109.0442|bibcode=2011PhRvD..84l2004R}}</ref>

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Suppose <math>Y</math> is a random vector with components <math>u,v</math> 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:

:<math>f(u,v) = {1\over{2\pi\sigma^{2}}}\left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}</math>

Let <math>R = \sqrt{U^{2}+V^{2}}</math> be the magnitude of <math>Y</math>. Then the [[cumulative distribution function]] (CDF) of the magnitude is:

:<math> F(r) = {1\over{2\pi\sigma^{2}}}\iint_{D_{r}} \left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}du \; dv </math>

where <math>D_{r}</math> is the disk defined by:

:<math> D_{r} = \left\{ (u,v) : \sqrt{u^{2}+v^{2}} \leq r \right\} </math>

Converting to [[polar coordinates]] leads to the CDF becoming:

:<math> \begin{aligned} F(r) &= {1\over{2\pi\sigma^{2}}}\int_{0}^{r}\int_{0}^{2\pi} \rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}d\theta \; d\rho \\ &= {1\over{\sigma^{2}}}\int_{0}^{r}\rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} d\rho \\ &= 1-\left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2}  \end{aligned} </math>

Finally, the [[probability density function]] (PDF) of the magnitude may be derived:

:<math> f(r) = F'(r) = {r\over{\sigma^{2}}} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} </math>

In the limit as <math> \nu \rightarrow \infty </math>, the Rayleigh distribution is recovered because:

:<math> \lim_{\nu\rightarrow \infty} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} = e^{-r^{2}/2\sigma^{2}} </math>

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