= Distribution sensitivity of randomized zero-endpoint probabilities
<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 $\widetilde p_0$, choose a positive upper endpoint $\widetilde u$ with nominal <cumulative distribution function> $\widetilde F(\widetilde u)=b$, and put $r=1-a/\widetilde p_0$, with $0<a\le\widetilde p_0<b<1$. 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 is
$$
C=F(\widetilde u)-p_0+rp_0
=F(\widetilde u)-a\frac{p_0}{\widetilde p_0}.
$$
Subtracting the target $b-a$ separates the two model errors:
$$
C-(b-a)=\{F(\widetilde u)-b\}
-\frac{a}{\widetilde p_0}(p_0-\widetilde p_0).
$$
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.<SciPy 1.16.2 documentation>[scipy.stats.uniform, support and rvs; scipy.stats.norm, cdf and ppf]{ref}
For an original misspecification example, choose $a=0.025$, $b=0.975$, and $\widetilde p_0=0.20$, so $r=0.875$. Suppose instead $p_0=0.30$ and $F(\widetilde u)=0.970$. Then
$$
C=0.970-0.125(0.30)=0.9325,
$$
which 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 $|F(\widetilde u)-b|\le\epsilon_F$ and $|p_0-\widetilde p_0|\le\epsilon_p$ imply $|C-(b-a)|\le\epsilon_F+(a/\widetilde p_0)\epsilon_p$. 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.
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