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Encyclopedia > Raised cosine distribution
 Probability density function Cumulative distribution function Parameters $mu,$(real) $s>0,$(real) In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Support $x in [mu-s,mu+s],$ pdf $frac{1}{2s} left[1+cosleft(frac{x!-!mu}{s},piright)right],$ cdf $frac{1}{2}left[1!+!frac{x!-!mu}{s} !+!frac{1}{pi}sinleft(frac{x!-!mu}{s},piright)right]$ Mean $mu,$ Median $mu,$ Mode $mu,$ Variance $s^2left(frac{1}{3}-frac{2}{pi^2}right),$ Skewness $0,$ Kurtosis $frac{6(90-pi^4)}{5(pi^2-6)^2},$ Entropy mgf $frac{pi^2sinh(s t)}{st(pi^2+s^2 t^2)},e^{mu t}$ Char. func. $frac{pi^2sin(s t)}{st(pi^2-s^2 t^2)},e^{imu t}$

In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval [μ − s,μ + s]. The probability density function is In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... Probability theory is the mathematical study of probability. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

$f(x;mu,s)=frac{1}{2s} left[1+cosleft(frac{x!-!mu}{s},piright)right],$

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 < 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:-1...

$E(x^{2n})=frac{1}{2}int_{-1}^1 [1+cos(xpi)]x^{2n},dx$
$= frac{1}{n!+!1}+frac{1}{1!+!2n},_1F_2 left(n!+!frac{1}{2};frac{1}{2},n!+!frac{3}{2};frac{-pi^2}{4}right)$

where $,_1F_2$ is a hypergeometric function. In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

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