= Thermal-product input-distribution check
A \b[thermal-product input-distribution check] tests whether the probability model used for a nonlinear thermal correction describes the available temperature and material information. In $d=La\theta$, with coefficient difference $a$ and temperature offset $\theta$, the uncertainty depends on their joint behaviour, not just on two separately quoted standard uncertainties.
Start with the physical inputs: which thermometer calibration is shared, how coefficient estimates were obtained, and whether either quantity was inferred using the other. Translate those relationships into a <joint distribution> or an explicit dependence model. JCGM 101 distinguishes assigning input distributions from propagating them through a <measurement model>; numerical sampling cannot supply missing physical information.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.2 and 6.1]{ref}
An original counterexample shows why zero <covariance> is insufficient for the independent-input formula. Let $Z$ be standard normal and define
$$
a=\sigma_aZ,\qquad \theta=\frac{\sigma_\theta}{\sqrt2}(Z^2-1).
$$
Both inputs have zero mean, with variances $\sigma_a^2$ and $\sigma_\theta^2$, and their covariance is zero because the required Gaussian odd moments vanish. They are nevertheless dependent: $\theta$ is determined by $a$.
Their product has zero mean and variance
$$
\operatorname{Var}(a\theta)=\frac{\sigma_a^2\sigma_\theta^2}{2}\mathbb E[Z^2(Z^2-1)^2]
=5\sigma_a^2\sigma_\theta^2,
$$
using Gaussian moments $\mathbb E[Z^2]=1$, $\mathbb E[Z^4]=3$, $\mathbb E[Z^6]=15$. Independence would instead give $\sigma_a^2\sigma_\theta^2$. With $L=100\,\mathrm{mm}$, $\sigma_a=10^{-6}\,\mathrm{K^{-1}}$ and $\sigma_\theta=0.1\,\mathrm K$, the resulting standard deviation is about 22.36 nm rather than 10 nm. This is a deliberately constructed probability example, not a claim about typical thermometers.
<Thermal-product dependence beyond covariance> matters because higher moments enter the product variance. A <multivariate normal distribution> is a particular joint model whose moments are determined by its means and covariance; arbitrary inputs sharing those same summaries need not behave identically.
Check units, support, marginal shapes and dependence before simulation. Compare exact or limiting cases with a <Monte Carlo method>, and retain a sensitivity analysis for plausible alternative input models. The GUM's first-order rule requires higher-order treatment when nonlinearity matters.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2]{ref} A small simulation error measures numerical precision conditional on the model, not proof that the model is physically correct.
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