is the sum of all the terms where . Such sums occur when investigating sums of random variables, and discrete versions appear in the coefficients of products of polynomials and power series. Convolution is an important tool in data processing, in particular in digital signal and image processing. We will first define the concept in various general settings, discuss its properties and then list several convolutions of probability distributions.
If is a locally compact (topological) Abelian group (http://planetmath.org/LocallyCompactGroupoids) with Haar measure and and are measurable functions on , we define the convolution
The case is the most important one, but is also useful, since it recovers the convolution of sequences which occurs when computing the coefficients of a product of polynomials or power series. The case yields the so-called cyclic convolution which is often discussed in connection with the discrete Fourier transform. Based on this definition one also obtains the groupoid C*–convolution algebra (http://planetmath.org/GroupoidCConvolutionAlgebra)
The (Dirichlet) convolution of multiplicative functions considered in number theory does not quite fit the above definition, since there the functions are defined on a commutative monoid (the natural numbers under multiplication) rather than on an abelian group.
If and are independent random variables with probability densities and respectively, and if has a probability density, then this density is given by the convolution . This motivates the following definition: for probability distributions and on , the convolution is the probability distribution on given by
for every Borel set . The last equation is the result of Fubini’s theorem.
The convolution of two distributions and on is defined by
for any test function for , assuming that is a suitable test function for .
which provides a great simplification in the computation of convolution. Because of the availability of the Fast Fourier Transform and its inverse, this latter relation is often used to quickly compute discrete convolutions, and in fact the fastest known algorithms for the multiplication of numbers and polynomials are based on this idea.
Some convolutions of probability distributions
The convolution of two distributions with and degrees of freedom is a distribution with degrees of freedom.
and the normal distribution has zero mean and variance , then for the probability density of the sum is
In a semi-logarithmic diagram where is plotted versus and versus , the latter lies by the amount higher than the former but both are represented by parallel straight lines, the slope of which is determined by the parameter .
The convolution of a uniform and a normal distribution results in a quasi-uniform distribution smeared out at its edges. If the original distribution is uniform in the region and vanishes elsewhere and the normal distribution has zero mean and variance , the probability density of the sum is
Adapted with permission from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
|Date of creation||2013-03-22 12:32:45|
|Last modified on||2013-03-22 12:32:45|
|Last modified by||PrimeFan (13766)|