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Im wrong about something here, but Im not sure what.

As far as I know the product of two normal distributed variables is not normal distributed. However, if the joint distribution of Y and X is bivariate normal then Y given X is normal distributed as well as the marginal distribution of X. Furthermore, the joint distribution of Y and X is the product of the condtional distribution of Y given X and the marginal distribution of X which are both normal. If we define a new variable Z to be distributed as the condional distribution of Y given X, then the product of Z and X (which are both normal distributed) is bivariate normal.

Am I on the right track or did I miss understand something somewhere?

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You're talking about two different things:

$$V=XY$$

vs.

$$f_{XY}(x,y)$$

The product of two variables cannot be bivariate, since a product is a number.

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