4
$\begingroup$

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

$\endgroup$

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, Najib Idrissi, TZakrevskiy, Davide Giraudo, Mark Fantini
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ 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
5
$\begingroup$

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 $n$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.