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In probability theory, a convolution is a mathematical operation that allows us 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, the convolution is obtained by summing a series of products of the probability mass functions (pmfs) of the two variables.

In the case of continuous random variables, it is obtained by integrating the product of their probability density functions (pdfs).

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 by using one of the following two formulae:[eq4]

These two summations are called convolutions.

Example Let X be a discrete variable with support [eq5] and pmf[eq6]Let Y be another discrete variable, independent of X, with support [eq7] and pmf[eq8]Their sum[eq9]has support [eq10]The pmf of Z needs to be calculated for every $zin R_{Z}$. For $z=0$, we have[eq11]For $z=1$, we get[eq12]For $z=2$, the pmf is[eq13]And so on, until we obtain the value of [eq3] for all $zin R_{Z}$.

Convolution of probability density functions

If X and Y are continuous, independent, and have probability density functions [eq15] and [eq16] respectively, the convolution formulae become[eq17]

Example Let X be a continuous variable with support [eq18] and pdf[eq19]that is, X has an exponential distribution. Let Y be another continuous variable, independent of X, with support [eq20] and pdf[eq21]that is, X has a uniform distribution. Define [eq9]The support of Z is[eq23]When $zin R_{Z}$, the pdf of Z is[eq24]Therefore, the probability density function of Z is[eq25]

More details

A more detailed explanation of the concept of convolution and the proofs of the two convolution formulae can be found in the lecture entitled Sums of independent random variables.

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How to cite

Please cite as:

Taboga, Marco (2021). "Convolutions", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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