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We prove that, for every $n\\ge1$, whenever $\\mathbb{E} X_i\\le1$ for every $i$, $\\mathbb{P}\\{K(X)\\le\\alpha\\}\\le\\alpha$ for all $0\\le\\alpha\\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\\mathbb{E}X_i \\le 1$ for all $i$. 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Thomas","submitted_at":"2026-07-09T12:39:29Z","abstract_excerpt":"Let $X=(X_1,\\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\\ldots,D_n)\\sim\\operatorname{Dir}(1,\\ldots,1)$ be independent of $X$, and define $K(x)=\\mathbb{P}\\{\\sum_{i=1}^n x_iD_i\\le1\\}$. We prove that, for every $n\\ge1$, whenever $\\mathbb{E} X_i\\le1$ for every $i$, $\\mathbb{P}\\{K(X)\\le\\alpha\\}\\le\\alpha$ for all $0\\le\\alpha\\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\\mathbb{E}X_i \\le 1$ for all $i$. 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