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?


  • $\begingroup$ 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 Nicolas Apr 10 '13 at 18:45
  • $\begingroup$ 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? $\endgroup$ – Perdue Apr 10 '13 at 18:48
  • $\begingroup$ 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 Nicolas Apr 10 '13 at 18:53
  • $\begingroup$ Crap...alright, I guess I am back to the drawing board on this one then. $\endgroup$ – Perdue Apr 10 '13 at 19:00
  • $\begingroup$ 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 Nicolas Apr 10 '13 at 19:28

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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