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{{Distinguish|Rayleigh mixture distribution}}
He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.<br><br>I keep conjuring up images of him in 1983, trying to reignite the [http://Mondediplo.com/spip.php?page=recherche&recherche=passion passion]. <br>I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.<br><br>He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.<br><br>The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.<br><br>99                  &#9733;&#9733;&#9733;&#9733;&#9733;Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.<br><br>And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �<br>This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.<br><br>Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx� <br>The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�<br>�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!<br><br>She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.<br><br>The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and [http://Channing.com/ Channing] Tatum<br>But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.<br><br>In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.<br><br>Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.<br><br>Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.<br><br>WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES<br>The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals <br>With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. &#9733;&#9733;&#9733;&#9733;&#9733;<br><br>If you have any sort of concerns regarding where and just how to utilize clash of clans triche gemmes ([http://nouveauclashofclanstriche.blogspot.com/ describes it]), you can call us at the website.
 
{{Probability distribution|
  name      =Rayleigh|
  type      =density|
  pdf_image =[[Image:Rayleigh distributionPDF.svg|325px|Plot of the Rayleigh PDF]]<br /><small></small>|
  cdf_image  =[[Image:Rayleigh distributionCDF.svg|325px|Plot of the Rayleigh CDF]]<br /><small></small>|
  parameters =<math>\sigma>0\,</math>|
  support    =<math>x\in [0,+\infty)</math>|
  pdf        =<math>\frac{x}{\sigma^2} e^{-x^2/2\sigma^2}</math>|
  cdf        =<math>1 - e^{-x^2/2\sigma^2}</math>|
  mean      =<math>\sigma \sqrt{\frac{\pi}{2}}</math>|
  median    =<math>\sigma\sqrt{\ln(4)}\,</math>|
  mode      =<math>\sigma\,</math>|
  variance  =<math>\frac{4 - \pi}{2} \sigma^2</math>|
  skewness  =<math>\frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}</math>|
  kurtosis  =<math>-\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}</math>|
  entropy    =<math>1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}</math>|
  mgf        =<math>1+\sigma t\,e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}
\left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)</math>|
  char      =<math>1\!-\!\sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}\!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)</math>|
}}
 
In [[probability theory]] and [[statistics]], the '''Rayleigh distribution''' {{IPAc-en|ˈ|r|eɪ|l|i}} is a [[continuous probability distribution]] for positive-valued [[random variable]]s.
 
A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional [[Euclidean_vector#Vector_components|components]]. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed into its orthogonal 2-dimensional vector components. Assuming that the magnitudes of each component are [[uncorrelated]], [[Normal distribution|normally distributed]] with equal [[variance]], and zero [[mean]], then the overall wind speed ([[Euclidean vector|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 i.i.d. (independently and identically distributed) [[normal distribution|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 [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]].{{Citation needed|date=April 2013}}
 
==Definition==
The [[probability density function]] of the Rayleigh  distribution is<ref Name=PP>Papoulis, Athanasios; Pillai, S. (2001) ''Probability, Random Variables and Stochastic Processe''. ISBN 0073660116, ISBN 9780073660110 {{Page needed|date=April 2013}}</ref>
 
:<math>f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}, \quad x \geq 0,</math>
 
where <math>\sigma >0,</math> is the [[scale parameter]] of the distribution. The [[cumulative distribution function]] is<ref Name=PP/>
 
:<math>F(x) = 1 - e^{-x^2/2\sigma^2}</math>
 
for <math>x \in [0,\infty).</math>
 
==Relation to random vector lengths==
 
Consider the two-dimensional vector <math> Y = (U,V) </math> which has components that are Gaussian-distributed and independent. Then <math> f_U(u; \sigma) = \frac{e^{-u^2/2\sigma^2}}{\sqrt{2\pi\sigma^2}} </math>, and similarly for <math> f_V(v; \sigma) </math>.
 
Let <math> x </math> be the length of <math> Y </math>. It is distributed as
 
: <math>f(x; \sigma) =  \frac{1}{2\pi\sigma^2} \int_{-\infty}^\infty du \, \int_{-\infty}^\infty dv \, e^{-u^2/2\sigma^2} e^{-v^2/2\sigma^2} \delta(x-\sqrt{u^2+v^2}).</math>
 
By transforming to the [[polar coordinate system]] one has
 
: <math> f(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_0^{2\pi} \, d\phi \int_0^\infty dr \, \delta(r-x) r e^{-r^2/2\sigma^2}= \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.
 
==Properties==
 
The raw [[moment (mathematics)|moments]] are given by:
 
:<math>\mu_k = \sigma^k2^\frac{k}{2}\,\Gamma\left(1 + \frac{k}{2}\right)</math>
 
where <math>\Gamma(z)</math> is the [[Gamma function]].
 
The [[mean]] and [[variance]] of a Rayleigh [[random variable]] may be expressed as:
 
:<math>\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253 \sigma</math>
 
and
 
:<math>\textrm{var}(X) = \frac{4 - \pi}{2} \sigma^2 \approx 0.429 \sigma^2</math>
 
The mode is <math>\sigma </math> and the maximum pdf is
 
:<math> f_\text{max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-\frac{1}{2}} \approx \frac{1}{\sigma} 0.606</math>
 
The [[skewness]] is given by:
 
:<math>\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^\frac{3}{2}} \approx 0.631</math>
 
The excess [[kurtosis]] is given by:
 
:<math>\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245</math>
 
The [[characteristic function (probability theory)|characteristic function]] is given by:
 
:<math>\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\textrm{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]</math>
 
where <math>\operatorname{erfi}(z)</math> is the imaginary [[error function]]. The [[moment generating function]] is given by
 
:<math>
  M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}}
          \left[\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]</math>
 
where <math>\operatorname{erf}(z)</math> is the [[error function]].
 
===Differential entropy===
The [[differential entropy]] is given by{{Citation needed|date=April 2013}}
 
:<math>H = 1 + \ln\left(\frac{\sigma}{\sqrt{2}}\right) + \frac{\gamma}{2}</math>
 
where <math>\gamma</math> is the [[Euler–Mascheroni constant]].
 
== Parameter estimation ==
 
Given a sample of ''N'' [[independent and identically distributed]] Rayleigh random variables <math>x_i</math> with parameter <math>\sigma</math>,
 
:<math>\widehat{\sigma^2}\approx \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2</math> is an unbiased [[maximum likelihood]] estimate.
 
:<math>\hat{\sigma}\approx \!\,\sqrt{\frac{1}{2N}\sum_{i=1}^N x_i^2}</math> is a biased estimator that can be corrected via the formula
 
:<math>\sigma = \hat{\sigma} \frac {\Gamma(N)\sqrt{N}} {\Gamma(N + \frac {1} {2})} = \hat{\sigma} \frac {4^N N!(N-1)!\sqrt{N}} {(2N)!\sqrt{\pi}}</math><ref>[https://archive.org/details/jresv68Dn9p1005 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]</ref>
 
=== Confidence intervals ===
To find the (1&nbsp;&minus;&nbsp;''α'') confidence interval, first find <math>\chi_1^2, \ \chi_2^2</math> where:
:&nbsp; <math>Pr(\chi^2(2n) \leq \chi_1^2) = \alpha/2, \quad Pr(\chi^2(2n) \leq \chi_2^2) = 1 - \alpha/2</math>
then
:&nbsp; <math>\frac{2n\overline{x^2}}{\chi_2^2} \leq \widehat{\sigma}^2 \leq \frac{2n\overline{x^2}}{\chi_1^2}</math><ref>[http://nvlpubs.nist.gov/nistpubs/jres/66D/jresv66Dn2p167_A1b.pdf 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]</ref>
 
== Generating random variates ==
 
Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] in the interval <nowiki>(0,&nbsp;1)</nowiki>, then the variate
 
:<math>X=\sigma\sqrt{-2 \ln(U)}\,</math>
 
has a Rayleigh distribution with parameter <math>\sigma</math>. This is obtained by applying the [[inverse transform sampling]]-method.
 
==Related distributions==
 
*<math>R \sim \mathrm{Rayleigh}(\sigma)</math> is Rayleigh distributed if <math>R = \sqrt{X^2 + Y^2}</math>, where <math>X \sim N(0, \sigma^2)</math> and <math>Y \sim N(0, \sigma^2)</math> are independent [[Normal_distribution|normal random variables]].<ref>[http://home.kpn.nl/jhhogema1966/skeetn/ballist/sgs/sgs.htm#_Toc96439743 Hogema, Jeroen (2005) "Shot group statistics"]</ref> (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
 
*The [[chi distribution]] with ''v''&nbsp;=&nbsp;2 is equivalent to Rayleigh Distribution with&nbsp;''&sigma;''&nbsp;=&nbsp;1.  I.e., if <math>R \sim \mathrm{Rayleigh} (1)</math>, then <math>R^2</math> has a [[chi-squared distribution]] with parameter <math>N</math>, degrees of freedom, equal to two (''N''&nbsp;=&nbsp;2)
:: <math>[Q=R^2] \sim \chi^2(N)\ .</math>
 
*If <math>R \sim \mathrm{Rayleigh}(\sigma)</math>, then <math>\sum_{i=1}^N R_i^2</math> has a [[gamma distribution]] with parameters <math>N</math> and <math>\frac{1}{2\sigma^2}</math>
:: <math>\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma(N,\frac{1}{2\sigma^2}) .</math>
 
*The [[Rice distribution]] is a generalization of the Rayleigh distribution.
 
*The [[Weibull distribution]]  is a generalization of the Rayleigh distribution.  In this instance, parameter <math>\sigma</math> is related to the Weibull scale parameter <math>\lambda</math>: <math>\lambda = \sigma \sqrt{2} .</math>
 
*The [[Maxwell–Boltzmann distribution]] describes the magnitude of a normal vector in three dimensions.
 
*If <math>X</math> has an [[exponential distribution]] <math>X \sim \mathrm{Exponential}(\lambda)</math>, then <math>Y=\sqrt{2X\sigma^2\lambda} \sim \mathrm{Rayleigh}(\sigma) .</math>
 
== Applications ==
An application of the estimation of σ can be found in [[magnetic resonance imaging]] (MRI). As MRI images are recorded as [[complex numbers|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.<ref>Sijbers J., den Dekker A. J., Raman E. and Van Dyck D. (1999) "Parameter estimation from magnitude MR images", ''International Journal of Imaging Systems and Technology'', 10(2), 109&ndash;114</ref>
 
==See also==
*[[Rayleigh fading]]
*[[Rayleigh mixture distribution]]
 
{{More footnotes|date=April 2013}}
 
== References ==
 
{{reflist}}
 
{{ProbDistributions|continuous-semi-infinite}}
 
{{DEFAULTSORT:Rayleigh Distribution}}
[[Category:Continuous distributions]]
[[Category:Exponential family distributions]]
[[Category:Probability distributions]]

Latest revision as of 06:03, 7 August 2014

He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.

I keep conjuring up images of him in 1983, trying to reignite the passion.
I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.

He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.

The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.

99 ★★★★★Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.

And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �
This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.

Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx�
The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�
�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!

She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.

The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum
But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.

In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.

Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.

Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.

WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES
The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals
With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. ★★★★★

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