Convolutions

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
Convolution of probability density functions
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Convolution of probability mass functions

Let be a discrete random variable with support and probability mass function .

Let be another discrete random variable, independent of , with support $R_{Y}$ and probability mass function .

The probability mass function of the sum can be derived by using one of the following two formulae: [eq4]

These two summations are called convolutions.

Example Let be a discrete variable with support and pmf [eq6] Let be another discrete variable, independent of , with support and pmf [eq8] Their sumhas support The pmf of needs to be calculated for every $zin R_{Z}$ . For , we have [eq11] For , we get [eq12] For , the pmf is [eq13] And so on, until we obtain the value of for all $zin R_{Z}$ .

Convolution of probability density functions

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

Example Let be a continuous variable with support and pdf [eq19] that is, has an exponential distribution. Let be another continuous variable, independent of , with support and pdf [eq21] that is, has a uniform distribution. Define The support of isWhen $zin R_{Z}$ , the pdf of is [eq24] Therefore, the probability density function of 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. https://www.statlect.com/glossary/convolutions.