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Say I sample $N$ numbers from a mystery discrete distribution and add them all together. My goal is for the sum of these random variables to be uniformly distributed from $[a,b]$ and thus the distribution is paramaterized by $(N, a, b)$. Does such a distribution exist? I can imagine how to calculate the probability of each number being picked here, but I was wondering if this had a name or if anyone had formalized it (My GoogleFu is failing me). I mentioned the discrete case as it's relevant to the problem I'm trying to solve, but I'm curious about the continuous case as well.

Tom Yurek
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    Given $X_1, X_2 \sim$ Uniform and iid , $X_1 +X_2 \sim$ triangular distribution: – user158565 Dec 05 '18 at 06:46
  • Although the duplicate deals explicitly with the case $N=2,$ all larger $N$ are also addressed for the simple reason that any sum of $N\gt 2$ values can be written (in many ways) as the sum of two sub-sums. – whuber Dec 05 '18 at 14:47

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