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Suppose we have a collection of discrete probability mass functions with different ranges, all of which are from 0 to some positive integer. As a simple example, we might be rolling 3 6-sided dice, 1 10-sided die, and 2 20-sided dice. (We'll ignore the 0 in this case) But, it could be any set of weighted dice with an arbitrary range, most likely all unique.

How can we define a joint PMF that provides the probability distribution over the sum of the rolled values? The multinomial distribution seems to be limited to repeated samples from one PMF.

My intuition tells me solving this problem for a large collection of PMFs (100 or so) would be computationally demanding, and I expect a Monte Carlo solution would be more realistic. However, I still seek a formal definition that I can refer to, and ideally with a reference or terminology so I can explore related work that may apply.

Thank you for your time.

Jeff

Jeff
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    This seems extremely difficult to do theoretically, and I would do some sort of Monte Carlo solution for this problem if it were given to me. One approach to do this, in case you're curious, is to consider all of the variable characteristic functions, find the characteristic function of the sum, and apply inversion to get the PMF. However, I'm not familiar enough with inversion to do this problem or to even know if it is doable. – Clarinetist Jun 11 '15 at 01:16

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