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).

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

Let be another discrete random variable, independent of , with support and probability mass function .

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

These two summations are called convolutions.

Example Let be a discrete variable with support and pmfLet be another discrete variable, independent of , with support and pmfTheir sumhas support The pmf of needs to be calculated for every . For , we haveFor , we getFor , the pmf isAnd so on, until we obtain the value of for all .

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

Example Let be a continuous variable with support and pdfthat is, has an exponential distribution. Let be another continuous variable, independent of , with support and pdfthat is, has a uniform distribution. Define The support of isWhen , the pdf of isTherefore, the probability density function of is

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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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.

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