Suppose I have N sets of numbers (10 numbers per set) {a1, ....., a10}. I form a sum by taking one number at random from each set. SUM = num from set 1 +......+ num from set N. If I do this a large number of times I will generate a large number of different values for the SUM variable. How can I estimate the distribution of the SUM values (P10, P50, P90 etc.)?

  • $\begingroup$ First of all, you haven't defined a distribution, but I'm assuming that you're picking the number uniformly and independently. Second, what do you mean by "estimate the distribution of Sum"? What kind of answer are you looking for? $\endgroup$ May 11, 2015 at 14:23

1 Answer 1


Use a convolution.

For instance, write the generating function for each set, then multiply the generating functions; this gives you the generating function for the distribution of the sum.

This gives you the exact distribution (which is even better than an approximation!).

  • $\begingroup$ D.W. In order to find the exact distribution of the values of SUM would I not have to evaluate all possible combinations of the summed values/ $\endgroup$ May 12, 2015 at 15:06
  • $\begingroup$ @DavidWilkinson, no, you would not. My answer describes how to do it, without evaluating all combinations. I recommend you read about convolutions and generating functions, if you're not already familiar with the subject, as they solve exactly this problem in an elegant, efficient way. $\endgroup$
    – D.W.
    May 13, 2015 at 1:41

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