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.
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 using one of the following two formulae:
These two summations are called convolutions.
If and are absolutely continuous and have probability density functions and , the convolution formulae become:
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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