# Is the joint distribution of $(XY,XZ)$ identified from distribution of product $XY$ and $XZ$ when $X,Y$ and $Z$ are iid random variables?

Let $$X,Y$$ and $$Z$$ be three independent and identically distributed random variables with unknown marginal distribution. If we know the distribution of product $$XY$$ and the distribution of product $$XZ$$ (essentially, $$XY$$ and $$XZ$$ should have the same distribution), then is the joint distribution of $$(XY,XZ)$$ also identified?

• The conditions could be stated more clearly. Is this correct: 1. $X,Y,Z$ are iid, but the common distribution $F$ is unknown? 2. But the common distribution of the products $XY, XZ$ are known, marginally? 3. Then the joint is also known/identified? Obstacle could be, basically, that different distributions $F,G$ give rise to the same product distribution, marginally, but then the joints differ? I guess this could be related to the Hamburger Moment Problem. If thats correct, some conditions might be needed. – kjetil b halvorsen Sep 9 at 11:06
• Thanks for your comments. I have edited the question. Hope it is clear now. I will check what you suggest. – zxjroger Sep 9 at 14:18