Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian

The Product of Two Gaussian Random Variables is not Gaussian distributed:

But the product of two Gaussian PDFs is a Gaussian PDF:

What is going on here?

What am I doing when I take the product of two pdfs
vs. when I take the product of two variables from the pdfs?

When (what physical situation) is described by one,
and what by the other?
(I think a few real world examples would clear things up for me)

Answer

The product of the PDFs of two random variables X and Y will give the joint distribution of the vector-valued random variable (X,Y) in the case that X and Y are independent. Therefore, if X and Y are normally distributed independent random variables, the product of their PDFs is bivariate normal with zero correlation.

On the other hand, even in the case that X and Y are IID standard normal random variables, their product is not itself normal, as the links you provide show. The product of X and Y is a scalar-valued random variable, not a vector-valued one as in the above case.

Attribution
Source : Link , Question Author : Lyndon White , Answer Author : heropup

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