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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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA