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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 $q_p=\inf \{\ell :\mathrm{Pr}(L\le \ell )\ge p\}$ for $0<p<1$ givesEquivalently, $q_p=\tau \max (0,\mu +s{\Phi }^{-1}(p))$. 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.^[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.^[ref] Propagate any uncertainty in decay time or other envelope parameters jointly before applying this fixed-$\tau$ formula.

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 $L\ge 0$ 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 $a\le p_0<b$. Choose upper endpoint $u$ with $\mathrm{Pr}(L\le u)=b$.

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 $U\sim \mathrm{Uniform}(0,1)$ and reportThen the joint probability over both $L$ and the auxiliary random draw isThe independence assumption is essential to the multiplication $r{p}_0$.

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.^[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.