= Coverage probability
{title2=$\mathbb P_\theta(\theta\in C(Y))$}
The <coverage probability> is the repeated-sampling <probability> that a random <confidence interval> contains the fixed true <statistical parameter>. A finite-sample level-$1-\alpha$ procedure has <coverage probability> at least $1-\alpha$ at every admissible <statistical parameter>; exact nominal coverage means equality. An asymptotically calibrated procedure approaches the nominal value as <sample size> grows, under its stated assumptions. Higher-order <Edgeworth expansions> can quantify the discrepancy from nominal <coverage probability> when their regularity conditions hold.
Observed coverage in finitely many repetitions is itself random. For $m$ independent repetitions of the exact <normal mean confidence interval with known variance>, the count $K$ of covering <confidence intervals> has <binomial distribution> $\operatorname{Binomial}(m,0.95)$. At $m=40$, its <expectation> is $38$, but
$$
\mathbb P(K=39)=\binom{40}{39}0.95^{39}0.05\approx0.270552.
$$
Consequently, the 39 successes in the figure are compatible with exact 95% <coverage probability>. The guarantee follows from the <pivotal quantity>'s <probability distribution>; the finite observed fraction is an illustration. Also distinguish a <confidence interval> for a fixed <mean> from a <prediction interval> for a future observation: the latter must accommodate the new observation's variation as well as estimation uncertainty.
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