A Fully Polynomial-Time Approximation Scheme for Approximating a Sum of Random Variables

Given $n$ independent random variables $X_1, X_2, ..., X_n$ and an integer $C$, we study the fundamental problem of computing the probability that the sum $X=X_1+X_2+...+X_n$ is at most $C$. We assume that each random variable $X_i$ is implicitly given by an oracle which, given an input value $k$, returns the probability $X_i\leq k$. We give the first deterministic fully polynomial-time approximation scheme (FPTAS) to estimate the probability up to a relative error of $1\pm \epsilon$. Our algorithm is based on the idea developed for approximately counting knapsack solutions in [Gopalan et al. FOCS11].