$X$ and $Y$ are two independent and identically distributed random variables. Can $X+Y$ be uniformly distributed over interval $[0,1]$?


closed as off-topic by heropup, Najib Idrissi, TZakrevskiy, Davide Giraudo, Mark Fantini Oct 5 '14 at 9:20

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  • $\begingroup$ Uniformly distributed over a bounded interval? $\endgroup$ – user180040 Oct 5 '14 at 7:02
  • $\begingroup$ Thanks. Yes, I did mean uniformly distributed over a bounded interval. $\endgroup$ – Rajat Oct 7 '14 at 2:07

No. Such an $X$ would have to be take its values on a bounded interval, and so its characteristic function would be entire. The characteristic function of the sum of two iid random variables is the square of the characteristic function of one of them. The characteristic function of a uniform random variable (say on $[0,1]$) is $\dfrac{i(1-e^{it})}{t}$, which does not have an entire square root because of the simple zeros at $t = 2 \pi n$ for nonzero integer $n$.


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