= Quantiles of a clipped Gaussian acquisition duration
{title2=$q_p=\tau\max(0,\mu+s\Phi^{-1}(p))$}
\b[<Quantiles> of a clipped Gaussian acquisition duration] assign probability levels to a recording extension whose distribution has a point mass at zero. Under the model of <Gaussian input uncertainty at a clipped acquisition-time threshold>, let $X\sim N(\mu,s^2)$, $s>0$, and $L=\tau\max(0,X)$ for fixed $\tau>0$. Its zero mass is $p_0=\Phi(-\mu/s)$.
Using the lower generalized <quantile> $q_p=\inf\{\ell:\Pr(L\leq\ell)\geq p\}$ for $0<p<1$ gives
$$
q_p=\begin{cases}
0,&p\leq p_0,\\
\tau\bigl(\mu+s\Phi^{-1}(p)\bigr),&p>p_0.
\end{cases}
$$
Equivalently, $q_p=\tau\max(0,\mu+s\Phi^{-1}(p))$. The constant branch is essential: directly using a negative untruncated normal <quantile> would assign an impossible negative duration. SciPy's normal-distribution documentation identifies `ppf` as the inverse cumulative distribution function used for the Gaussian part of this calculation.<SciPy 1.16.2 documentation>[scipy.stats.norm, Methods: ppf, cdf and loc/scale parameters]{ref}
For an original threshold example, take $\mu=0$, $\tau=30\,\mathrm{ns}$ and $s=0.01$. Half the probability is at zero, so the median is zero. The 90th and 95th percentiles are approximately $0.3845$ and $0.4935\,\mathrm{ns}$. The one-sided interval $[0,q_{0.95}]$ has probability $0.95$ under this stated model.
A nominal equal-tail 95 percent construction behaves differently: $[q_{0.025},q_{0.975}]=[0,0.5880]\,\mathrm{ns}$ actually contains probability $0.975$. Its lower endpoint includes the entire zero atom instead of excluding a 2.5 percent lower tail. This <coverage excess from central quantiles at a zero-duration atom> is a property of the mixed distribution, not a rounding error.
<Quantiles> describe uncertainty in the conditional acquisition requirement. They are not automatically a frequentist <confidence set> for an unknown fixed physical duration, nor a worst-case guarantee on unrecorded optical energy. NIST explains why an interval's probability interpretation requires the underlying distributions and assumptions.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§6.3, Expanded Uncertainty]{ref} Propagate any uncertainty in decay time or other envelope parameters jointly before applying this fixed-$\tau$ formula.
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