## A Sum of Gaussian Random Variables is a Gaussian Random Variable

A basic result from the theory of random variables is that when you
*sum* two independent random variables, you *convolve* their
probability density functions (PDF). (Equivalently, in the frequency
domain, their *characteristic functions multiply*.)

That the sum of two independent Gaussian random variables is Gaussian follows immediately from the fact that Gaussians are closed under multiplication (or convolution).

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