AI 中文总结
该论文通过自包含的证明，确立了关于独立非负随机变量和的概率下界的公式，证明了Samuels猜想，且由此一并解决了Feige猜想。
AI 中文摘要
设0≤μ₁≤…≤μₙ，且λ>∑ᵢ₌₁ⁿμᵢ；设X₁,…,Xₙ为独立非负随机变量，满足EXᵢ=μᵢ，记Dᵢ:=λ−∑ₖ₌₁ⁱ⁻¹μₖ（1≤i≤n）。我们证明：inf_{X₁,…,Xₙ}P(∑ᵢ₌₁ⁿXᵢ<λ)=min_{1≤i≤n}∏ⱼ₌ᵢⁿ(1−μⱼ/Dᵢ)，该界是尖锐的且可达，由此证明了Samuels猜想；因Feige猜想可由等均值情形直接推出，故也一并解决了Feige猜想，证明是自包含的。
英文摘要
Let $0\leqμ_1\leq\cdots\leqμ_n$ and let $λ>\sum_{i=1}^nμ_i$. Let $X_1,...,X_n$ be independent nonnegative random variables satisfying $\mathbb{E}X_i=μ_i$, and write $D_i := λ-\sum_{k=1}^{i-1}μ_k$ for $1\leq i\leq n$. We prove that $$ \inf_{X_1,...,X_n} \mathbb{P}\left( \sum_{i=1}^nX_i<λ\right) = \min_{1\leq i\leq n} \prod_{j=i}^n \left( 1-\frac{μ_j}{D_i} \right). $$ The bound is sharp and is attained. This proves Samuels' conjecture. Feige's conjecture is thereby resolved, since it follows immediately from the equal-means case. The proof is self-contained.
