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by Marco Taboga, PhD

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.

Table of Contents

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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