Convolution of Two Densities

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The convolution of two functions can be thought of as a measure of the overlap of the graphs as one graph is shifted horizontally across the other. Formally, if and are functions, the convolution of the two is the function .

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The plot shows , that is, shifted by units, in blue, in purple, and the product of the two in gold. Thus the gray area is exactly the value of the convolution at .

If and are independent random variables with respective density functions and , then the density function of is the convolution of and . Interestingly, the convolution of two Gaussian densities is a Gaussian density.

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Contributed by: Chris Boucher (March 2011)
Open content licensed under CC BY-NC-SA


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