When one adds two uniform random distributions, the sum forms a so called Irwin-Hall distribution. This is described well in the litteratur.
My problem is the opposite one: I want to find two (identical) random distributions that are such that when I add them together they form a uniform random distribution.
Through a brute-force numerical search I have been able find an approximate distribution that fulfills this. However, I was hoping that someone could point me to a more mathematical/elegant approach.
https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution
If you want independent random variables (which one would assume), the answer is that this cannot be done. For a proof, please see http://stats.stackexchange.com/questions/125360 . This is also an exercise in Grimmet & Stirzaker, Probability and Random Processes.
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