Raised cosine distribution
Encyclopedia
In probability theory
and statistics
, the raised cosine distribution is a probability distribution
supported on the interval [
]. The probability density function is
for
and zero otherwise. The cumulative distribution function is
for
and zero for 
and unity 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
and 
. Because the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:
where
is a generalized hypergeometric function.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the raised cosine distribution is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
supported on the interval [
]. The probability density function isfor
and zero otherwise. The cumulative distribution function isfor
and zero for and unity for .The moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
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
and . Because the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:where
is a generalized hypergeometric function.See also
- Hann functionHann functionThe Hann function, named after the Austrian meteorologist Julius von Hann, is a discrete probability mass function given byw= 0.5\; \leftorw= \sin^2 \left...