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Convolution of Two Densities

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 .
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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