# 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]?

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 $U[0,1]$.