Distribution of Sum of Discrete Uniform Random Variables?

Question

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!

Answer 1

A normal distribution is a continuous distribution, which we can never get by summing discrete random variables. The sum of discrete random variables will be another discrete random variable. Also, it depends on how many discrete uniform random variables you're adding together. If you add 2 independent identical discrete uniform random variables, the result will be a discrete random variable with a triangular profile. Add 3 and you get a discrete random variable with a parabolic profile, and so on.

The Central Limit Theorem tells us that a sum of many continuous random variables will approach a gaussian. Similarly, for discrete random variables, the profile of such a sum will have a gaussian shape, though the actual distribution will approach a binomial distribution.