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$$X = \sum_i^n x_i$$

,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.

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There is an article on this for sum of non-IID uniforms. I haven't read the article through, but seems relevant. ON THE DISTRIBUTION OF THE SUM OF n NON-IDENTICALLY DISTRIBUTED UNIFORM RANDOM VARIABLES Oh, also theorem 2.2 here – user1705135 May 15 '14 at 2:27
up vote 7 down vote accepted

The sum of $n$ iid random variables with (continuous) uniform distribution on $[0,1]$ has distribution called the Irwin-Hall distribution. Some details about the distribution, including the cdf, can be found at the above link. One can then get corresponding information for uniforms on $]a,b]$ by linear transformation.

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What is linear transfer? What if the uniforms are not identically distributed? – Matt Munson Apr 23 '13 at 14:42
I meant linear transformation: $Y$ is uniform om $[a,b]$ iff $\frac{Y-a}{b-a}$ is uniform on $[0,1]$. If not iid, distribution doesn't have a name that I know, which is no problem. But the cdf, even for modestly small $n$, is extremely messy. I once needed it for $8$, and there were some symmetries, but I had to work very hard. This was before good symbolic manipulators. – André Nicolas Apr 23 '13 at 15:22
Ok, so the distribution when not iid is nameless as far as we know, but then how do I find it? Should I just try to use the convolution as the other poster suggests? – Matt Munson Apr 23 '13 at 15:48
The convolution process works in principle, it always does. You will find that evaluating the integral is unpleasant at $n=2$, and a real challenge at $n=3$. The real question is: what do you need to know about the sum? Maybe you can get it out of the moment generating function. – André Nicolas Apr 23 '13 at 15:53
The Irwin-Hall referral is not strictly appropriate, because the OP specifies: $x_i \in (a_i,b_i)$ ... i.e. that the domain of support for each $x_i$ varies with i. This is different to the Irwin-Hall set-up which assumes that the domain of support is the same for all $x_i$. – wolfies Apr 26 '13 at 13:09

The PDF of $X$ is given by the convolution of the PDFs of the variables $x_i$.

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