On Samuels' Conjecture

Let $0\leq\mu_1\leq\cdots\leq\mu_n$ and let $\lambda>\sum_{i=1}^n\mu_i$. Let $X_1,...,X_n$ be independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$, and write $D_i := \lambda-\sum_{k=1}^{i-1}\mu_k$ for $1\leq i\leq n$. We prove that $$ \inf_{X_1,...,X_n} \mathbb{P}\left( \sum_{i=1}^nX_i<\lambda \right) = \min_{1\leq i\leq n} \prod_{j=i}^n \left( 1-\frac{\mu_j}{D_i} \right). $$ 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.