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The sample minimum and maximum enclose random population content given by the uniformized sample range. The exact failure probability for content at least $1-\epsilon$ is $(1-\epsilon {)}^{n-1}[1+(n-1)\epsilon ]$. For small $\epsilon$, setting $n\epsilon \approx \vartheta$ gives the displayed sample-size approximation at confidence $1-\delta$. The positive solution is unique for $0<\delta <1$, since $(1+\vartheta )e^{-\vartheta }$ decreases from one to zero. A tolerance interval concerns population content, rather than a confidence interval for a fixed parameter; the exact inequality should select the final integer size.

For $n\ge 2$, the probability integral transform makes $U_i=F(X_i)$ iid uniform on $[0,1]$. Continuity of $F$ is enough; strict increase is unnecessary. Their extreme order statistics have joint densitychoose the observations giving the two extremes and put the remaining $n-2$ between them. Transform to $r=v-u$ and integrate $u$ from zero to $1-r$. The uniformized sample range therefore hasEquivalently $R\sim \mathrm{Beta}(n-1,2)$. For $n=1$, the range is identically zero instead.

The exact failure probability isThus the exact integer sample size is the least $n\ge 2$ making this quantity at most $\delta$. For $\epsilon \downarrow 0$ with $n\epsilon \to \vartheta >0$, it tends to $(1+\vartheta )e^{-\vartheta }$. This function decreases strictly from one to zero on positive $\vartheta$, because its derivative is $-\vartheta e^{-\vartheta }$. For $0<\delta <1$ there is therefore exactly one positive solution ofThis gives the extreme-order-statistic tolerance interval sample-size ruleWhen $\delta$ is small, $\vartheta =\log (1/\delta )+\log (1+\vartheta )$, so a useful first refinement is $\vartheta \approx \log (1/\delta )+\log \log (1/\delta )$. This is an approximation, not an exact integer guarantee; use the preceding exact failure probability to check rounding and very stringent tail requirements. The interval here encloses population content, rather than estimating a fixed distribution parameter.