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Suppose that I have $n$ independent Bernoulli variables $b_i$ with their associated probability of success $p_i$, and a set of corresponding weights $w_i$.

I would like to say something about the distribution of the random variable defined as: $$ x=\sum_{i=1}^nw_ib_i$$ Is this a known distribution? Are there any useful results about this?

Leonardo Massai
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    See Section 3 of https://arxiv.org/abs/1107.2702. "They're hard to learn from samples." (By the way, if all $w_i$'s are equal to 1, this is called a Poisson Binomial Distribution.) – Clement C. Dec 19 '19 at 15:15
  • @ClementC. Presumably from the usage of the term "weights" we should assume that the $x$ is a convex combination of the $b_i$, i.e. $w_i\geqslant 0$ and $\sum_{i=1}^n w_i=1$. – Math1000 Dec 19 '19 at 20:21
  • https://math.stackexchange.com/questions/1546366/distribution-of-weighted-sum-of-bernoulli-rvs –  Dec 19 '19 at 20:25
  • @Math1000 The linked paper assumes the weights are known, so the two are equivalent for the hardness result. You can always renormalize your draws. – Clement C. Dec 20 '19 at 05:53

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