Convolution of probability distributions: Difference between revisions

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The general formula for the distribution of the sum <math>Z=X+Y</math> of two independent integer-valued (and hence discrete) random variables is<ref>[[Susan P. Holmes|Susan Holmes ]](1998). Sums of Random Variables:

Statistics 116. Stanford. https://web.archive.org/web/20210413200454/http://statweb.stanford.edu/~susan/courses/s116/node114.html</ref>

:<math>P(Z=z) = \sum_{k=-\infty}^\infty P(X=k)P(Y=z-k)</math>

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There are several ways of deriving formulae for the convolution of probability distributions. Often the manipulation of integrals can be avoided by use of some type of [[generating function]]. Such methods can also be useful in deriving properties of the resulting distribution, such as moments, even if an explicit formula for the distribution itself cannot be derived.

One of the straightforward techniques is to use [[Characteristic function (probability theory)|characteristic functions]], which always exists and are unique to a given distribution.{{citation needed|date=April 2013}}. It is possible to prove convolution as the sum of multiple dice rolling<ref>[[The Sum of Outcomes for Rolling an N-sided Dice K Times Through Convolution ]](2024). https://github.com/dipuk0506/dice</ref>.

=== Convolution of Bernoulli distributions ===