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I have thirty discrete random variables for a risk management application. Each of the random variables may have a value $0$ (with probability $p_1$) or $X_i>0$ (with probability $p_2$). All thirty $X_i$ may be different. I am interested in the sum $S=\sum_{i=1}^{30} X_i$ of all thirty random variables. Am I right regarding the following thoughts:

1) an exact analytic solution for the probability distribution of S would demand the computation of $2^{30}$ different value-probability-pairs

2) the expected value of S can easily be calculated as the sum of 30 expected values

3) the standard deviation of S can only be calculated easily, if all 30 have a mutual covariance of 0

4) for a risk management application, which must consider values as well as probabilities, we must apply a Monte-Carlo-approch because of 1)-3) as there is no quick algorithm for an analytic solution

5) all would be even more clearly demanding the Monte-Carlo-Approach as in 4), if we have three differing instead of two differing values for the 30 random variables

Before posting the question, I checked

https://math.stackexchange.com/questions/295363/sum-of-two-random-variables

which only speaks about identical discrete distributions. Othe questions I saw on cross-validated seem to address special discrete distributions like Bernoulli or continuous distributions like the normal Distribution.

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1) Yes, can $S$ have $2^{30}$ distinct values, each with its own probability.

2) Yes.

3) Yes, as $Var(\sum X_i)=\sum_i \sum_j Cov(X_i,X_j)$

4) That depends on your specific question. You need some kind of tail probability? The dependence structure of the $X_i$ is really far from independence?

Something different would be to take a single distribution $F$ and model the $X_i$ as at most 30 non independent replicates from it (e.g. according to some copula). Then take a discrete distribution on $\{1,\ldots,30\}$. It's a different setting, but you can probably find more literature about it.

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