Say I have $n$ independent Bernoulli random variables, with parameters $p_1,\ldots,p_n$. Say, also, that I wish to decide whether their sum exceeds some given threshold $t$ with probability at least $0.5$. What is the computational complexity of this decision problem, when $p_1,\ldots,p_n$ and $t$ are represented in binary and given as input?
More generally, I'm interested in the generalization of this problem to (non-Bernoulli) discrete distributions. Specifically, there are $n$ independent random variables, each supported on at most $m$ rational numbers, with each variable's probability histogram given explicitly in the input. In this case, also, I want to decide whether the sum of these variables exceeds $t$ with probability at least $0.5$.
I have a feeling this problem is PP-hard, though I can't quite prove it. I wonder what the answer is, and whether it's already known.
Note that I'm not looking for approximation algorithms for this problem. It's clear that monte carlo methods yield positive answers to approximate versions of this decision problem. I'm interested in the exact decision problem as stated above.