Raised cosine distribution

Probability distribution
name =Raised cosine
type =density
pdf_

|
cdf_


parameters =$mu,$(real)
$s>0,$(real)
support =$x\; in\; [mu-s,mu+s]\; ,$
pdf =$frac\{1\}\{2s\}left\; [1+cosleft(frac\{x!-!mu\}\{s\},pi\; ight)\; ight]\; ,$
cdf =$frac\{1\}\{2\}left\; [1!+!frac\{x!-!mu\}\{s\}!+!frac\{1\}\{pi\}sinleft(frac\{x!-!mu\}\{s\},pi\; ight)\; ight]$
mean =$mu,$
median =$mu,$
mode =$mu,$
variance =$s^2left(frac\{1\}\{3\}-frac\{2\}\{pi^2\}\; ight),$
skewness =$0,$
kurtosis =$frac\{6(90-pi^4)\}\{5(pi^2-6)^2\},$
entropy =
mgf =$frac\{pi^2sinh(s\; t)\}\{st(pi^2+s^2\; t^2)\},e^\{mu\; t\}$
char =$frac\{pi^2sin(s\; t)\}\{st(pi^2-s^2\; t^2)\},e^\{imu\; t\}$
In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval [$mu-s,mu+s$] . The probability density function is

:$f(x;mu,s)=frac\{1\}\{2s\}left\; [1+cosleft(frac\{x!-!mu\}\{s\},pi\; ight)\; ight]\; ,$

for $mu-s\; le\; x\; le\; mu+s$ and zero otherwise. The cumulative distribution function is

:$F(x;mu,s)=frac\{1\}\{2\}left\; [1!+!frac\{x!-!mu\}\{s\}!+!frac\{1\}\{pi\}sinleft(frac\{x!-!mu\}\{s\},pi\; ight)\; ight]$

for $mu-s\; le\; x\; le\; mu+s$ and zero for

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 $s=1$. Since the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:

:$E(x^\{2n\})=frac\{1\}\{2\}int\_\{-1\}^1\; [1+cos(xpi)]\; x^\{2n\},dx$:$=\; frac\{1\}\{n!+!1\}+frac\{1\}\{1!+!2n\},\_1F\_2left(n!+!frac\{1\}\{2\};frac\{1\}\{2\},n!+!frac\{3\}\{2\};frac\{-pi^2\}\{4\}\; ight)$

where $,\_1F\_2$ is a hypergeometric function.