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

TL;DR

This paper presents the first deterministic fully polynomial-time approximation scheme (FPTAS) for estimating the probability that the sum of independent random variables is at most a given value, with applications in probabilistic analysis.

Contribution

The paper introduces a novel FPTAS for computing the probability of a sum of independent random variables being below a threshold, improving computational efficiency.

Findings

First deterministic FPTAS for this problem

Achieves relative error of 1±ε in polynomial time

Builds on techniques from knapsack solution counting

Abstract

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].