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I know that the product of two Gaussian functions is also Gaussian function. This is stated in Wikipedia, but I might need to cite a classical (text)book/paper stating this result. A book containing the proof is welcome. Could you name such a book?

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This is really just a simple consequence of well-kown identities concerning powers. – Rasmus Aug 28 '10 at 12:16
@Rasmus: Thanks, I didn't think before asking :|. I maintain the question only because of @John's answer. – lmsasu Sep 7 '10 at 7:28
up vote 4 down vote accepted

The product of two independent normal (Gaussian) random variables is not normal. That would be convenient, but it's not true. Here are a some related theorems that are true:

  • The sum of two independent normal random variables is normal.

  • The product of two independent log normal random variables is log normal.

  • The ratio of two independent normal random variables is Cauchy.

For more on distribution relationships, see this chart.

If you multiply two normal PDFs you get the joint PDF of a multivariate normal, i.e. if f(x) is a normal PDF then g(x, y) defined by f(x)g(y) is a multivariate normal PDF, but in that case you're multiplying densities and not random variables.

Update: I looked back at your question and I see you said Gaussian "function," rather than "random variable", so perhaps you had in mind the PDF result I mentioned above. If so, just write down the definition of multivariate normal and there it is.

Update 2: Here's an explicit formula for the the product of two normal pdfs being proportional to another normal pdf: blog post.

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Unfortunately John's blog post doesn't contain the full derivation. Would anyone like to contribute a line-by-line proof? Or cite a textbook that has one? – Neil Traft Sep 12 '14 at 20:13

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