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| {{Probability distribution|
| | Hello and welcome. My name is Irwin and I totally dig that title. My family members lives in Minnesota and my family loves it. What I love performing is performing ceramics but I haven't made a dime with it. For years he's been working as a receptionist.<br><br>Feel free to visit my web blog at home std testing ([http://www.animecontent.com/blog/349119 click through the following website page]) |
| name =Raised cosine|
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| type =density|
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| pdf_image =[[Image:RCosine distribution PDF.png|325px|Plot of the raised cosine PDF]]<br /><small></small>||
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| cdf_image =[[Image:RCosine distribution CDF.png|325px|Plot of the raised cosine CDF]]<br /><small></small>|
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| parameters =<math>\mu\,</math>([[real number|real]])<br>
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| <math>s>0\,</math>([[real number|real]])|
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| support =<math>x \in [\mu-s,\mu+s]\,</math>|
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| pdf =<math>\frac{1}{2s}
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| \left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,</math>|
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| cdf =<math>\frac{1}{2}\left[1\!+\!\frac{x\!-\!\mu}{s}
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| \!+\!\frac{1}{\pi}\sin\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]</math>|
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| mean =<math>\mu\,</math>|
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| median =<math>\mu\,</math>|
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| mode =<math>\mu\,</math>|
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| variance =<math>s^2\left(\frac{1}{3}-\frac{2}{\pi^2}\right)\,</math>|
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| skewness =<math>0\,</math>|
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| kurtosis =<math>\frac{6(90-\pi^4)}{5(\pi^2-6)^2}\,</math>|
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| entropy =|
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| mgf =<math>\frac{\pi^2\sinh(s t)}{st(\pi^2+s^2 t^2)}\,e^{\mu t}</math>|
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| char =<math>\frac{\pi^2\sin(s t)}{st(\pi^2-s^2 t^2)}\,e^{i\mu t}</math>|
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| }}
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| In [[probability theory]] and [[statistics]], the '''raised cosine distribution''' is a continuous [[probability distribution]] supported on the interval <math>[\mu-s,\mu+s]</math>. The [[probability density function]] is
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| :<math>f(x;\mu,s)=\frac{1}{2s}
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| \left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,</math>
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| for <math>\mu-s \le x \le \mu+s</math> and zero otherwise. The cumulative distribution function is
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| :<math>F(x;\mu,s)=\frac{1}{2}\left[1\!+\!\frac{x\!-\!\mu}{s}
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| \!+\!\frac{1}{\pi}\sin\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]</math>
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| for <math>\mu-s \le x \le \mu+s</math> and zero for <math>x<\mu-s</math> and unity for <math>x>\mu+s</math>.
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| The [[moment (mathematics)|moments]] of the raised cosine distribution are somewhat complicated, but are considerably simplified for the standard raised cosine distribution. The standard raised cosine distribution is just the raised cosine distribution with <math>\mu=0</math> and <math>s=1</math>. Because the standard raised cosine distribution is an [[Even and odd functions|even function]], the odd moments are zero. The even moments are given by:
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| :<math>E(x^{2n})=\frac{1}{2}\int_{-1}^1 [1+\cos(x\pi)]x^{2n}\,dx </math>
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| :<math>= \frac{1}{n\!+\!1}+\frac{1}{1\!+\!2n}\,_1F_2
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| \left(n\!+\!\frac{1}{2};\frac{1}{2},n\!+\!\frac{3}{2};\frac{-\pi^2}{4}\right)</math>
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| where <math>\,_1F_2</math> is a [[generalized hypergeometric function]].
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| ==See also==
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| * [[Hann function]]
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| == References ==
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| *{{Cite web
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| | author = Horst Rinne
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| | url = http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf
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| | title = Location-Scale Distributions - Linear Estimation and Probability Plotting Using MATLAB
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| | year = 2010
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| | page = 116
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| | accessdate = 2012-11-16
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| }}
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| {{ProbDistributions|continuous-bounded}}
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| {{DEFAULTSORT:Raised Cosine Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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Hello and welcome. My name is Irwin and I totally dig that title. My family members lives in Minnesota and my family loves it. What I love performing is performing ceramics but I haven't made a dime with it. For years he's been working as a receptionist.
Feel free to visit my web blog at home std testing (click through the following website page)