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Given a random variable $$X = \sum_i^n x_i,$$ where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$?

Ariel Serranoni
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Matt Munson
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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 http://arxiv.org/pdf/math/0411298v1.pdf Oh, also theorem 2.2 here http://www.heldermann-verlag.de/eqc/eqc01_16/eqc16002.pdf – user1705135 May 15 '14 at 02:27

2 Answers2

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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.

bjschoenfeld
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André Nicolas
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  • What is linear transfer? What if the uniforms are not identically distributed? – Matt Munson Apr 23 '13 at 14:42
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    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
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    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
  • Various moments and other statistics that one needs in practical work can be found without explicit formulas for the distribution. Such formulas, if ugly enough, just get in the way. – André Nicolas Apr 23 '13 at 17:30
  • Cool. Of course the problem here is that I have to learn moments. They don't exactly seem trivial either. – Matt Munson Apr 24 '13 at 00:52
  • If the "real" problem can be stated fairly simply, probablists can (probably) solve it. – André Nicolas Apr 24 '13 at 01:08
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    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
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The PDF of $X$ is given by the convolution of the PDFs of the variables $x_i$.

Matt L.
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