Convolution of probability distributions: Difference between revisions

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==Introduction==

The [[probability distribution]] of the sum of two or more [[independent (probability)|independent]] [[random variable]]s is the convolution of their individual distributions. The term is motivated by the fact that the [[probability mass function]] or [[probability density function]] of a sum of independent random variables is the [[convolution]] of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions: see [[List of convolutions of probability distributions]].

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>

For independent, continuous random variables with pdf[[probability density function]]s (PDF) <math>f,g</math> and [[cumulative distribution function]]s (CDF) <math>F,G</math> respectively, we have that the CDF of the sum is:

:<math>H(z)=\int_{-\infty}^\infty F(z-t)g(t) dt = \int_{-\infty}^\infty G(t)f(z-t) dt</math>