I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution?
Thanks!
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$\begingroup$ First we want to specify that the discrete uniforms are independent and over the same interval. The sum is not normal, but the sum of a largish number of them is close enough to normal for most practical uses. $\endgroup$– André NicolasCommented Apr 10, 2013 at 18:45
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$\begingroup$ So if I have a sample of n discrete uniforms, that would be close enough to normal for the purposes of finding a complete statistic? $\endgroup$– PerdueCommented Apr 10, 2013 at 18:48
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$\begingroup$ If you look up the defintion of complete statistic, you will find that the answer is no. In particular, for complete statistic the thing would have to woek for a sample of, say, $3$, where normal approximation is really not good. $\endgroup$– André NicolasCommented Apr 10, 2013 at 18:53
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$\begingroup$ Crap...alright, I guess I am back to the drawing board on this one then. $\endgroup$– PerdueCommented Apr 10, 2013 at 19:00
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$\begingroup$ I don;t have time to give an answer, but for example if we are looking at the continuous uniform family in the interval $[0,\theta]$ where $\theta$ is a parameter, then the maximum of the $n$ observations is a complete statistic. This should be even more true for a similar family, discrete uniform on the interval $[0,\theta]$ where the parameter $\theta$ is an integer. $\endgroup$– André NicolasCommented Apr 10, 2013 at 19:28
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2 Answers
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The distribution is asymptotically normal. Otherwise, the exact distribution is that of a normalized extended binomial coefficient, see "Polynomial Coefficients and Distribution of the Sum of Discrete Uniform Variables" by Camila C. S. Caiado and Pushpa N. Rathie at http://community.dur.ac.uk/c.c.d.s.caiado/multinomial.pdf.
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2$\begingroup$ This explanation seems to be more approachable: dartmouth.edu/~chance/teaching_aids/books_articles/… $\endgroup$ Commented Feb 10, 2017 at 4:24
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$\begingroup$ For later reference (in case of 404 errors), the URL to the full book Introduction to Probability by Charles M. Grinstead and J. Laurie Snell is dartmouth.edu/~chance/teaching_aids/books_articles/… $\endgroup$ Commented May 12, 2017 at 5:29
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$\begingroup$ Hi Casey and Ludovic, Thanks for your information. I can find the paper written by Camila and Pushpa. But I still get confused about Polynomial Coefficient, expecially how to calculate it. May I ask how to calculate the Polynomial Coefficient by using software, say R or Matlab? $\endgroup$– BinCommented Sep 29, 2021 at 14:25
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So, the idea is to iteratively calculate PMF, using sum of $n-1$ variables to derive PMF of the sum of $n$ variables.
$$P_{a, b, n}(x) = \sum_{i \in a..b} P_{a, b, 1}(i) * P_{a, b, n-1}(x-i)$$
where $P_{a,b,n}$ is the PMF of the sum of $n$ discrete uniform variables.
Notional Python code to calculate the coefficients can be found here.
