Raised cosine distribution: Information from Answers.com

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Raised cosine
Probability density function
Cumulative distribution function
parameters:	$\mu\,$ (real) $s>0\,$ (real)
support:	$x \in [\mu-s,\mu+s]\,$
pdf:	$\frac{1}{2s} \left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,$
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:	$0\,$
kurtosis:	$\frac{6(90-\pi^4)}{5(\pi^2-6)^2}\,$
entropy:
mgf:	$\frac{\pi^2\sinh(s t)}{st(\pi^2+s^2 t^2)}\,e^{\mu t}$
cf:	$\frac{\pi^2\sin(s t)}{st(\pi^2-s^2 t^2)}\,e^{i\mu t}$

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.

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell–Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

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Raised cosine distribution

See also