= Coverage excess from central quantiles at a zero-duration atom
{title2=$\Pr(q_a\leq L\leq q_b)=F(q_b)-F(q_a^-)$}
\b[Coverage excess from central <quantiles> at a zero-duration atom] is the additional distributional probability included when a closed <quantile> interval captures an entire point mass at zero. Here coverage means probability under the stated duration model. For $L=\tau\max(0,X)$ with $X\sim N(\mu,s^2)$, the zero mass is $p_0=\Phi(-\mu/s)$ and the positive part is continuous. Let $a=\alpha/2$, $b=1-\alpha/2$, and use the lower <quantiles> from <quantiles of a clipped Gaussian acquisition duration>.
For the closed central interval $[q_a,q_b]$, its probability is
$$
\Pr(q_a\leq L\leq q_b)=\begin{cases}
1-\alpha,&p_0<a,\\
b,&a\leq p_0\leq b,\\
p_0,&p_0>b.
\end{cases}
$$
To verify this, write the probability as $F(q_b)-F(q_a^-)$. When both endpoints are positive, the lower left limit is $a$ and the upper value is $b$. Once the lower endpoint is zero, its left limit is zero; if the upper endpoint is zero too, the interval contains exactly the zero atom. Endpoint inclusion is therefore part of the probability calculation.
For a nominal 95 percent central construction, zero masses $0.01$, $0.20$ and $0.99$ give actual probabilities $0.95$, $0.975$ and $0.99$ respectively. In the middle case, the excess is $0.025$, not the whole $0.20$ atom: the interval already includes the probability between zero and the upper <quantile>.
As an original dimensional example, set $\tau=20\,\mathrm{ns}$, $s=0.02$ and $\mu/s\approx0.84162$, giving $p_0=0.20$. The nominal central 95 percent interval is approximately $[0,1.1206]\,\mathrm{ns}$ and contains probability $0.975$. SciPy documents the normal cdf and inverse-cdf ppf used to evaluate these Gaussian-component values.<SciPy 1.16.2 documentation>[scipy.stats.norm, Methods: cdf and ppf]{ref}
A one-sided upper interval or a separately specified probability construction can target a different included probability. <Randomized endpoint inclusion at a zero-duration atom> would require an explicit randomization rule, not a change made silently after seeing an endpoint equal zero. NIST explains why interval interpretation requires the input distributions and the chosen construction.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§6.3, Expanded Uncertainty]{ref} Distributional probability for this conditional acquisition model is separate from a worst-case optical-energy guarantee or a confidence procedure for an unknown fixed parameter.
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