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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3This is really just a simple consequence of well-kown identities concerning powers. – Rasmus Aug 28 '10 at 12:16
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@Rasmus: Thanks, I didn't think before asking :|. I maintain the question only because of @John's answer. – lmsasu Sep 07 '10 at 07:28
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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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John D. Cook
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1Unfortunately 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