 Quantiles of a clipped Gaussian acquisition duration

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.