= Randomized endpoint inclusion at a zero-duration atom
{title2=$r=1-\alpha/(2p_0)$}
\b[Randomized endpoint inclusion at a zero-duration atom] adds an independent random decision about whether an interval contains the atom, allowing an exact included probability under a specified distribution. Let the duration $L\geq0$ have zero mass $p_0$ and a continuous positive part. For a nominal central level $1-\alpha$, put $a=\alpha/2$, $b=1-\alpha/2$ and assume $a\leq p_0<b$. Choose upper endpoint $u$ with $\Pr(L\leq u)=b$.
The <coverage excess from central quantiles at a zero-duration atom> shows that the closed set $[0,u]$ has probability $b$, while $(0,u]$ has probability $b-p_0$. Draw an independent $U\sim\operatorname{Uniform}(0,1)$ and report
$$
I_U=(0,u]\ \cup\ \begin{cases}\{0\},&U\leq r,\\\varnothing,&U>r,\end{cases}
\qquad r=1-\frac a{p_0}.
$$
Then the joint probability over both $L$ and the auxiliary random draw is
$$
\Pr(L\in I_U)=b-p_0+rp_0=b-a=1-\alpha.
$$
The independence assumption is essential to the multiplication $rp_0$.
For an original example, let $p_0=0.20$ and $\alpha=0.05$. The endpoint is included with probability $r=0.875$. The positive part contributes $0.775$ probability and randomized zero inclusion contributes $0.175$, totaling $0.95$. An individual reported set is still either closed, with duration-model probability $0.975$, or open at zero, with probability $0.775$. Exact $0.95$ applies to the randomized procedure jointly, not conditionally to each realized set.
The <distribution sensitivity of randomized zero-endpoint probabilities> matters if $p_0$ or the upper endpoint is estimated inaccurately. Plugging a guessed atom probability into $r$ does not establish the same equality for the true distribution. The pinned numerical reference documents a standard <uniform distribution> and its random-variates method, while its normal cdf/ppf methods evaluate the Gaussian component when the duration comes from a clipped normal input.<SciPy 1.16.2 documentation>[scipy.stats.uniform, support and rvs; scipy.stats.norm, cdf and ppf]{ref}
This is a mathematical probability construction. It does not remove the atom physically, improve measurement precision, or create a worst-case guarantee on missed optical energy. Its randomization rule and probability interpretation must be specified as part of the procedure.
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