 Solution

An idealized underlying probability space iswith the product Borel sigma-algebra. Its 27 independent random coordinates are $\mu$, $\sigma$, twenty past standard-normal noises, and five future standard-normal noises. Define $T_i(\omega )=\mu +\sigma {\epsilon }_i$. Sampling an unknown variance does not mean drawing a new variance on each day; a single draw of ${\sigma }^2$ applies to all durations in a trace.

Let $A$ be the event that the first twenty durations satisfy their corresponding observed inequalities. The observations are the twenty late/early labels, conditional on the fixed departure thresholds. They are not exact values of the latent $T_i$. The posterior probability measure on traces isHere $\mathbb{P}(A)>0$: at every interior parameter pair each strict normal-tail probability lies in $(0,1)$, so their finite product has positive prior integral.

The returned outcome isThus the underlying sample space has dimension 27, the successful output space is $\{0,1\}$, and the observations define the conditioning event $A$. The desired output probability distribution is the pushforward measure of ${\mathbb{P}}_A$ under $Y$. Rejection is an implementation device, not a third scientific answer: None is discarded before forming the conditional output law.