Raised cosine distribution
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Probability density function |
|
Cumulative distribution function |
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| Parameters |  (real)
|
|---|---|
| Support | ![x \in [\mu-s,\mu+s]\,](../../../../math/a/6/f/a6f9fe4f12eb7b09726308e1e974302d.png) |
| Probability density function (pdf) | ![\frac{1}{2s}
\left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,](../../../../math/c/3/5/c357dc61c374de0f4f5dfe7e926e0722.png) |
| Cumulative distribution function (cdf) | ![\frac{1}{2}\left[1\!+\!\frac{x\!-\!\mu}{s}
\!+\!\frac{1}{\pi}\sin\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]](../../../../math/8/9/6/8966fcadb5917df590f683ee4e9ca0b7.png) |
| Mean | |
| Median | |
| Mode | |
| Variance |  |
| Skewness |  |
| Excess kurtosis |  |
| Entropy | |
| Moment-generating function (mgf) |  |
| Characteristic function |  |
(In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval [μ − s,μ + s]. The probability density function is
![f(x;\mu,s)=\frac{1}{2s}
\left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,](../../../../math/1/6/8/168c78e0f05ac1a575554370651d6d31.png)
for 
and zero otherwise. The cumulative distribution function is
![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/a/8/5/a858e6b6913f5863456015565f26f395.png)
for and zero for x < μ − s and unity for x > μ + s.
The 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 μ = 0 and s = 1. Since the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:
![E(x^{2n})=\frac{1}{2}\int_{-1}^1 [1+\cos(x\pi)]x^{2n}\,dx](../../../../math/b/b/8/bb808a6e11004eef1c075d94116d84d2.png)
where 
is a hypergeometric function.
