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The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
The Poisson summation formula is a particular case of the convolution theorem on tempered distributions. If one of the two factors is the Dirac comb , one obtains periodic summation on one side and sampling on the other side of the equation.
When is a periodic summation of another function, , then is known as a circular or cyclic convolution of and . And if the periodic summation above is replaced by f T {\displaystyle f_{T}} , the operation is called a periodic convolution of f T {\displaystyle f_{T}} and g T {\displaystyle g_{T}} .
The convolution theorem states that: [1] [2]: eq.8 ... is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, ...
In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density ...
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
More generally, given a monoid S, one can form the semigroup algebra [] of S, with the multiplication given by convolution. If one takes, for example, S = N d {\displaystyle S=\mathbb {N} ^{d}} , then the multiplication on C [ S ] {\displaystyle \mathbb {C} [S]} is a generalization of the Cauchy product to higher dimension.
The formula is also correct if f and g are functions from the positive integers into some abelian group (viewed as a Z-module). In the language of Dirichlet convolutions, the first formula may be written as = where ∗ denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. The second formula is then written as