Raised cosine distribution

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Raised cosine distribution

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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	$0\,$
Ex. kurtosis	$\frac{6(90-\pi^4)}{5(\pi^2-6)^2}\,$
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 continuous probability distribution supported on the interval [ $\mu-s,\mu+s$ ]. 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 $x<\mu-s$ and unity for $x>\mu+s$ .

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 $\mu=0$ and $s=1$ . Because 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 generalized hypergeometric function.

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

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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