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Let $\{X_i\}_{i=1}^N$ be $N$ random variables uniformly distributed over the intervals $[a_i, b_i]$ respectively. How does the sum:

$$\sum_{i=1}^N X_i$$

distribute?

This is a generalization of the Irwin-Hall distribution, which applies when all the intervals $[a_i,b_i]$ are equal.

becko
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    See e.g. "On the distribution of the sum of independent uniform random variables" by S. M. Sadooghi-Alvandi · A. R. Nematollahi · R. Habibi, Statistical Papers 2009 – Christoph Hanck Apr 14 '16 at 12:43
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    You could use just about any the many techniques I describe at http://stats.stackexchange.com/a/43075/919. It will be messy, but at the very least you could first assume all the $a_i$ are zero by subtracting their sum from $\sum X_i$, thereby reducing the problem to one where the support of each $X_i$ is $[0, b_i-a_i]$. – whuber Apr 14 '16 at 16:03

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