In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval ![{\displaystyle [\mu -s,\mu +s]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4092914759a1beeec30258141ccc43c0686f8459)
. The probability density function (PDF) is
Raised cosine|
Probability density function 
|
|
Cumulative distribution function 
|
| Parameters |
 (real)
 (real) |
|---|
| Support |
![{\displaystyle x\in [\mu -s,\mu +s]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021cb61824dc30c9ce4228710410d45d7b8ea2dd) |
|---|
| 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) |
|---|
| 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) |
|---|
| Mean |
|
|---|
| Median |
|
|---|
| Mode |
|
|---|
| Variance |
 |
|---|
| Skewness |
 |
|---|
| Excess kurtosis |
 |
|---|
| MGF |
 |
|---|
| CF |
 |
|---|
![{\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){\text{ for }}\mu -s\leq x\leq \mu +s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e881d815068ad8b253740178c997fe2d569289)
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.
