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This question occurred when thinking about the thundering herd problem so I could somehow generate random delays to make load on a server more uniform instead of a big spike when a large number of requests is generated at the same time.

Is there an uniformly distributed $X$ such that

$$ X = \sum_i^n x_i $$

and what is the distribution of iid $x_i$ if it exists?

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    $\begingroup$ Actually you are asking wheter the uniform distribution is infinitely divisible en.wikipedia.org/wiki/Infinite_divisibility_(probability) , to which the answer is no. $\endgroup$ Apr 29, 2020 at 11:21
  • $\begingroup$ If the question is "does there exist any n for which X is a sum of n iid variables?" then the answer "no" does not follow from the fact that the uniform distribution is not infinitely divisible. $\endgroup$
    – Valentas
    Nov 3, 2022 at 16:04

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Leander Tilsted Kristensen posted an answer in a comment. Thanks!

https://en.wikipedia.org/wiki/Infinite_divisibility_(probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed random variables.

...

The uniform distribution and the binomial distribution are not infinitely divisible (...)

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