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Given

$$X = \sum_i^n x_i$$

,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 arxiv.org/pdf/math/0411298v1.pdf Oh, also theorem 2.2 here heldermann-verlag.de/eqc/eqc01_16/eqc16002.pdf –  user1705135 May 15 at 2:27

2 Answers 2

up vote 5 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
    
Good point. I need to split the distribution at the mean and then find the mean of the half of the distribution. For each $x_i$ in the sum there will also be $x_j = -x_i$, so I believe the distribution will be symmetric and have a mean of $0$. But I need the mean of only the density on one side of the mean. Being able to calculate the mean for other arbitrary partitions would be useful also. –  Matt Munson Apr 23 '13 at 17:24

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

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