For i.i.d. random varianbles XX, YY, can X-YX-Y be uniform [0,1]?

Is there any distribution for two i.i.d. random variables X,Y where the joint distribution of X-Y is uniform over support [0,1]?

Answer

No.

If Y is ever (with positive probability) > X, then X – Y < 0, so it can't be U[0,1]. If X and Y are iid, Y can not be guaranteed (i.e., with probability 1) to not be > X unless X and Y are both the same constants with probability 1. In such case X - Y will equal 0 with probability 1. Therefore, there exists no iid X and Y such that X - Y is U[0,1].

Attribution
Source : Link , Question Author : Desmarais , Answer Author : Mark L. Stone

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