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This is a simple linear equation: $\mathbf{M}*A=\mathbf{S}$

M is a square matrix whose values are only 0 or 1.

A is the unknown vector

S is a data vector.

There are many random samples of M ans S, and is necessary to find the statistical distribution of each value in A, or at least his average values.

Expressed in other way, I have many random samples of s, but each sample of s is a sum of different combinations of $a_i$

I don't know how to solve it. I would just calculate $A_i=\mathbf{M_i}^{-1}*\mathbf{S_i}$, and just make a statistic of each value of A, but that may be a naive approach with wrong results.

I guess that this is a common problem with known solutions, but don't know what the name of the problem is, and where I should look for the solution.

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This is multiple linear regression. https://en.wikipedia.org/wiki/General_linear_model You're trying to estimate the regression coefficients $A$ in your question. There is a lot of material on this subject.

edit: Wrote $S$ instead of $A$.

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    No is not a regression at all. The variables $a_i$ don't need to have any correlation at all, and it doesn't matter if they are correlated. – 2017-02-08
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    s is not the unknown. A is the unknown. S is the data. – 2017-02-08
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    That was a typo on my part, swap S for A. Search multiple linear regression, you'll most likely find what you're looking for. – 2017-02-09