{{Short description|Probability distribution}} {{Probability distribution| name =Raised cosine| type =density| pdf_image =325px|Plot of the raised cosine PDF<br />|| cdf_image =325px|Plot of the raised cosine CDF<br />| parameters =<math>\mu\,</math>(real)<br> <math>s>0\,</math>(real)| support =<math>x \in [\mu-s,\mu+s]\,</math>| pdf =<math>\frac{1}{2s} \left[1+\cos\left(\frac{x-\mu}{s}\,\pi\right)\right]\,=\frac{1}{s}\operatorname{hvc}\left(\frac{x-\mu}{s}\,\pi\right)\,</math>| cdf =<math>\frac{1}{2}\left[1+\frac{x-\mu}{s} +\frac{1}{\pi}\sin\left(\frac{x-\mu}{s}\,\pi\right)\right]</math>| mean =<math>\mu\,</math>| median =<math>\mu\,</math>| mode =<math>\mu\,</math>| variance =<math>s^2\left(\frac{1}{3}-\frac{2}{\pi^2}\right)\,</math>| skewness =<math>0\,</math>| kurtosis =<math>\frac{6(90-\pi^4)}{5(\pi^2-6)^2}=-0.59376\ldots\,</math>| entropy =| mgf =<math>\frac{\pi^2\sinh(s t)}{st(\pi^2+s^2 t^2)}\,e^{\mu t}</math>| char =<math>\frac{\pi^2\sin(s t)}{st(\pi^2-s^2 t^2)}\,e^{i\mu t}</math>| }} 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 (PDF) is
<math display="block">\begin{align} f(x;\mu,s) &=\frac{1}{2s} \left[1+\cos\left(\frac{x-\mu}{s}\,\pi\right)\right]\\ &= \frac{1}{s}\operatorname{hvc}\left(\frac{x-\mu}{s}\,\pi\right) & \text{ for } \mu-s\le x\le\mu+s \end{align} </math>
and zero otherwise. The cumulative distribution function (CDF) is
<math display="block">F(x;\mu,s)=\frac{1}{2}\left[1+\frac{x-\mu}{s} + \frac{1}{\pi} \sin\left(\frac{x-\mu}{s} \, \pi \right) \right]</math>
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>.
The moments of the raised cosine distribution are somewhat complicated in the general case, 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 function, the odd moments are zero. The even moments are given by:
<math display="block"> \begin{align} \operatorname E(x^{2n}) & = \frac{1}{2}\int_{-1}^1 [1+\cos(x\pi)]x^{2n}\,dx = \int_{-1}^1 x^{2n} \operatorname{hvc}(x\pi)\,dx \\[5pt] & = \frac{1}{n+1}+\frac{1}{1+2n}\,_1F_2 \left(n+\frac{1}{2}; \frac{1}{2}, n+\frac{3}{2}; \frac{-\pi^2}{4} \right) \end{align} </math>
where <math>\,_1F_2</math> is a generalized hypergeometric function.
== See also == * Hann function * Havercosine (hvc)
== References == *{{Cite web | author = Horst Rinne | url = http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf | title = Location-Scale Distributions – Linear Estimation and Probability Plotting Using MATLAB | year = 2010 | page = 116 | access-date = 2012-11-16 }}
{{ProbDistributions|continuous-bounded}}
{{DEFAULTSORT:Raised Cosine Distribution}} Category:Continuous distributions