 Thermal-product input-distribution check

A 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.^[ref]

An original counterexample shows why zero covariance is insufficient for the independent-input formula. Let $Z$ be standard normal and defineBoth 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 varianceusing 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.^[ref] A small simulation error measures numerical precision conditional on the model, not proof that the model is physically correct.