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"i.i.d" has not been defined in this article, so I wrote "independent identically distributed" |
Added general convolution formula. |
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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 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 discrete variables is <ref>Susan Holmes (1998). Sums of Random Variables:
Statistics 116. Stanford. 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>:
The counterpart for independent continous variables with density functions <math>f(x), g(y)</math> is
:<math>h(z)=(f*g)(z)=\int_{-\infty}^\infty f(z-t)g(t) dt = \int_{-\infty}^\infty f(t)g(z-t) dt.</math>
== Example derivation ==
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