Raised cosine distribution

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Raised cosine
Probability density function
Plot of the raised cosine PDF
Cumulative distribution function
Plot of the raised cosine CDF
Parameters \mu\,(real)

s>0\,(real)
Support x \in [\mu-s,\mu+s]\,
Probability density function (pdf) \frac{1}{2s} \left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,
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]
Mean \mu\,
Median \mu\,
Mode \mu\,
Variance s^2\left(\frac{1}{3}-\frac{2}{\pi^2}\right)\,
Skewness 0\,
Excess kurtosis \frac{6(90-\pi^4)}{5(\pi^2-6)^2}\,
Entropy
Moment-generating function (mgf) \frac{\pi^2\sinh(s t)}{st(\pi^2+s^2 t^2)}\,e^{\mu t}
Characteristic function \frac{\pi^2\sin(s t)}{st(\pi^2-s^2 t^2)}\,e^{i\mu t}

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]\,

for \mu-s \le x \le \mu+s 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]

for \mu-s \le x \le \mu+s 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
= \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)

where \,_1F_2 is a hypergeometric function.

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