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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?

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  • $\begingroup$ 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. $\endgroup$ – kjetil b halvorsen Sep 9 at 11:06
  • $\begingroup$ Thanks for your comments. I have edited the question. Hope it is clear now. I will check what you suggest. $\endgroup$ – zxjroger Sep 9 at 14:18

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