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Atom-probability error in a randomized endpoint rule is the change in included duration-model probability caused by using an inaccurate zero-atom probability in the randomization. Let $0<a<b<1$, and hold the positive upper endpoint $u$ fixed at an actual cumulative distribution function value $F(u)=b$. If the true zero probability is $p_0$ but the rule uses a positive ${\hat{p}}_0\ge a$, its independent endpoint-inclusion probability is $r=1-a/{\hat{p}}_0$. Conditional on that chosen value,This is the atom term isolated in distribution sensitivity of randomized zero-endpoint probabilities, with upper-endpoint error deliberately removed. Underestimating $p_0$ reduces inclusion; overestimating it increases inclusion. The independent auxiliary draw can be generated from a uniform distribution.^[ref]

If ${\hat{p}}_0$ comes from an independent calibration, conditional probabilities must also be averaged over that calibration. For an original example, take $p_0=0.20$, $a=0.025$, and $b=0.975$. Suppose the calibration returns $0.10$ or $0.30$ with equal probability. Its estimate is unbiased, yet the resulting included probabilities are $0.925$ and $0.95833$, averaging about $0.94167$ rather than $0.95$.

More generally, for an unbiased positive estimate in the valid rule range, Jensen's inequality givesThis reciprocal-estimation bias in randomized atom inclusion comes from a nonlinear plug-in rule, even with independent calibration. It does not identify the size of the error without a calibration model. Values below $a$, including zero, require an explicitly defined clipped or guarded rule; the displayed inverse formula cannot simply be used there.

If a validated upper bound $p_U\ge p_0$ is available and $p_U\ge a$, choosing $r=1-a/p_U$ instead guarantees $C\ge b-a$ under the fixed-upper-endpoint and independence assumptions. It may be conservative rather than exact. Uncertainty in $F(u)$ remains a separate error term, and these are probabilities for a duration distribution, not a removal of the physical zero-duration atom.

Distribution sensitivity of randomized zero-endpoint probabilities quantifies how the probability equality in randomized endpoint inclusion at a zero-duration atom changes when its assumed duration law is wrong. Let the nominal law have zero-atom probability ${\tilde{p}}_0$, choose a positive upper endpoint $\tilde{u}$ with nominal cumulative distribution function $\tilde{F}(\tilde{u})=b$, and put $r=1-a/{\tilde{p}}_0$, with $0<a\le {\tilde{p}}_0<b<1$. Use the parent's independently randomized decision to include zero.

If the actual zero probability is $p_0$ and the actual cumulative distribution function is $F$, the included probability isSubtracting the target $b-a$ separates the two model errors:The first term is upper-endpoint probability error; the second is atom-probability error in a randomized endpoint rule. This identity is an algebraic consequence of independence of the auxiliary random draw, not a normal-approximation result. The documented uniform distribution random-variates construction supplies that auxiliary draw; normal cdf/ppf routines can evaluate a specified clipped-normal duration model.^[ref]

For an original misspecification example, choose $a=0.025$, $b=0.975$, and ${\tilde{p}}_0=0.20$, so $r=0.875$. Suppose instead $p_0=0.30$ and $F(\tilde{u})=0.970$. Thenwhich is $0.0175$ below $0.95$: $0.005$ comes from the upper endpoint and $0.0125$ from the atom. These are stipulated distributions, not measured failure rates.

More generally, bounds $|F(\tilde{u})-b|\le {ϵ}_F$ and $|p_0-{\tilde{p}}_0|\le {ϵ}_p$ imply $|C-(b-a)|\le {ϵ}_F+(a/{\tilde{p}}_0){ϵ}_p$. This propagates validated model-error bounds only. Estimating the nominal law from the same observation used for the interval can add dependence; the unconditional fixed-rule identity then requires an appropriate joint or conditional argument.

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.