Raised cosine
Probability density function ![Plot of the raised cosine PDF](https://upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Raised_cos_pdf_mod.svg/325px-Raised_cos_pdf_mod.svg.png)
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Cumulative distribution function ![Plot of the raised cosine CDF](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Raised_cos_cdf_mod.svg/325px-Raised_cos_cdf_mod.svg.png)
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Parameters |
(real)
(real) |
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Support |
![{\displaystyle x\in [\mu -s,\mu +s]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021cb61824dc30c9ce4228710410d45d7b8ea2dd) |
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PDF |
![{\displaystyle {\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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8fe6565ff842d25cf9ac9946e3454f278992d8) |
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CDF |
![{\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5fe6b908cecf264d0bc4a34c554b027ad3bb88) |
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Mean |
![{\displaystyle \mu \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340) |
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Median |
![{\displaystyle \mu \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340) |
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Mode |
![{\displaystyle \mu \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d20addf0d9f04e185714134b97726c4bf17d340) |
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Variance |
![{\displaystyle s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7f0f404dd1a16c6b6da93f6ccaf6f14f5f62c7) |
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Skewness |
![{\displaystyle 0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5) |
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Ex. kurtosis |
![{\displaystyle {\frac {6(90-\pi ^{4})}{5(\pi ^{2}-6)^{2}}}=-0.59376\ldots \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc88424327bb477a7b7f2ccfc81770d203b9158) |
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MGF |
![{\displaystyle {\frac {\pi ^{2}\sinh(st)}{st(\pi ^{2}+s^{2}t^{2})}}\,e^{\mu t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d21e99194bdb13fed62051f9619e117a5351c655) |
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CF |
![{\displaystyle {\frac {\pi ^{2}\sin(st)}{st(\pi ^{2}-s^{2}t^{2})}}\,e^{i\mu t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/475dedce20184e622782027322f89b284c9d09ee) |
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In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval
. The probability density function (PDF) is
![{\displaystyle 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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1332aa518f41cc75f685cbc018050732d4971473)
for
and zero otherwise. The cumulative distribution function (CDF) is
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d946fb3c3452f89b48341393ced089a0699fdffd)
for
and zero for
and unity for
.
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
and
. Because the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:
![{\displaystyle {\begin{aligned}\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}}\,_{1}F_{2}\left(n+{\frac {1}{2}};{\frac {1}{2}},n+{\frac {3}{2}};{\frac {-\pi ^{2}}{4}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd88646853daa97101c07fa637ef17568602b698)
where
is a generalized hypergeometric function.
See also
References
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Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and singular | |
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Families | |
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