Extreme-order-statistic tolerance interval ($n\approx \vartheta /\epsilon ,(1+\vartheta )e^{-\vartheta }=\delta$)

Let $X_1,\dots ,X_n$ be independent and identically distributed observations from a continuous cumulative distribution function $F$, with $n\ge 2$. The interval from the sample minimum to maximum contains random population probability $C=F(X_{(n)})-F(X_{(1)})$. The probability integral transform identifies $C$ with the uniformized sample range, soFor small $\epsilon$, setting $n\epsilon \approx \vartheta$ makes this approximately $(1+\vartheta )e^{-\vartheta }$. Solve $(1+\vartheta )e^{-\vartheta }=\delta$ for the unique positive $\vartheta$ when $0<\delta <1$, then use the exact displayed inequality to choose an integer sample size.

This is a distribution-free tolerance interval for population content under the continuous iid model. It differs from a confidence interval for a fixed parameter. Atoms or dependent observations require their own treatment; uniformizing them does not automatically preserve the same ordinary sample-range calculation.