= Gaussian input uncertainty at a clipped acquisition-time threshold
{title2=$L=\tau\max(0,X),\quad X\sim N(\mu,s^2)$}
\b[Gaussian input uncertainty at a clipped acquisition-time threshold] produces a mixed output distribution rather than an ordinary Gaussian error bar. Hold the decay time $\tau>0$ fixed and assume the logarithmic input ratio $X=\log(A\tau/\varepsilon)$ has distribution $N(\mu,s^2)$, with $s>0$. The required extension is
$$
L=\tau\max(0,X).
$$
The <directional derivative of a clipped logarithmic acquisition time> explains the kink; here the full input distribution is propagated through it.
Writing $\Phi$ and $\varphi$ for the standard normal cumulative distribution function and density, respectively,
$$
\Pr(L=0)=\Phi(-\mu/s),\qquad
\Pr(L\leq\ell)=\Phi\left(\frac{\ell/\tau-\mu}{s}\right)\quad(\ell\geq0).
$$
For $\ell>0$ the continuous part has density $\varphi((\ell/\tau-\mu)/s)/(\tau s)$. It integrates to less than one because the remaining probability is an atom at zero. Treating this density alone as the entire distribution loses that atom.
At the nominal threshold $\mu=0$, half the outcomes are zero and the conditional positive part has a <half-normal distribution>. Direct integration of the normal density gives
$$
\mathbb E[L]=\frac{\tau s}{\sqrt{2\pi}},\qquad
\operatorname{Var}(L)=\tau^2s^2\left(\frac12-\frac1{2\pi}\right).
$$
For an original example, $\tau=20\,\mathrm{ns}$ and logarithmic <standard deviation> $s=0.02$ give mean extension about $0.1596\,\mathrm{ns}$ and <standard deviation> $0.2335\,\mathrm{ns}$, although inserting the nominal input into the clipped formula gives zero. The probability of an extension above $0.80\,\mathrm{ns}$ is $1-\Phi(2)\approx0.0228$.
The <quantiles of a clipped Gaussian acquisition duration> must likewise include the zero atom. A symmetric interval around the nominal zero duration cannot represent these probabilities. NIST's uncertainty discussion stresses the need to know and combine input distributions when assigning an interval probability.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§6.3, Expanded Uncertainty]{ref}
This is a stated uncertainty model for a conditional recording requirement, not a worst-case optical-energy guarantee. Uncertain decay time, correlation among the original envelope parameters or a non-Gaussian logarithmic ratio requires the corresponding joint propagation instead. Establishing the optical envelope remains a separate experimental task.
Back to article page
