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In probability theory, convolution is a mathematical operation that allows to derive the distribution of a sum of two random variables from the distributions of the two summands. In the case of discrete random variables, it involves summing a series of products of their probability mass functions. In the case of continuous random variables, it is obtained by integrating the product of their probability density functions.

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Convolution of probability mass functions

Let X be a discrete random variable with support R_X and probability mass function [eq1]. Let Y be another discrete random variable, independent of X, with support $R_{Y}$ and probability mass function [eq2]. The probability mass function [eq3] of the sum $Z=X+Y$ can be derived using one of the following two formulae:[eq4]

These two summations are called convolutions.

Convolution of probability density functions

If X and Y are absolutely continuous and have probability density functions [eq5] and [eq6], the convolution formulae become:[eq7]

More details

A more detailed explanation of the concept of convolution - as well as some examples - can be found in the lecture entitled Sums of independent random variables.

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