Raised Cosine Distribution
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High Quality Content by WIKIPEDIA articles! In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval [ s, + s]. The probability density function is f(x;mu,s)=frac{1}{2s} left[1+cosleft(frac{x!-!mu}{s},piright)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}sinleft(frac{x!-!mu}{s},piright)right]for mu-s le x le mu+s and zero for x + s.The moments of the raised cosine distribution are somewhat complicated, but are considerably si...
High Quality Content by WIKIPEDIA articles! In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval [ s, + s]. The probability density function is f(x;mu,s)=frac{1}{2s} left[1+cosleft(frac{x!-!mu}{s},piright)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}sinleft(frac{x!-!mu}{s},piright)right]for mu-s le x le mu+s and zero for x + 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 = 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(xpi)]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.
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