= Thermal-product dependence beyond covariance
\b[Thermal-product dependence beyond covariance] is the influence of higher joint moments on uncertainty in a nonlinear <thermal expansion> correction. Two input models can have identical means, variances and <covariance> yet predict different uncertainty in a product. A <thermal-product input-distribution check> must therefore examine how the temperature offset and expansion-coefficient error occur together.
Let $a=\mu_a+A$ and $\theta=\mu_\theta+T$, where the centered variables have <expected values> zero, variances $s_a^2,s_\theta^2$ and covariance $c=\mathbb E[AT]$. Direct expansion gives
$$
\begin{aligned}
\mathbb E[a\theta]&=\mu_a\mu_\theta+c,\\
\operatorname{Var}(a\theta)&=\mu_a^2s_\theta^2+\mu_\theta^2s_a^2
+2\mu_a\mu_\theta c\\
&\quad+2\mu_a\mathbb E[AT^2]+2\mu_\theta\mathbb E[A^2T]
+\mathbb E[A^2T^2]-c^2.
\end{aligned}
$$
The third joint moments and the <mixed fourth moment of thermal inputs> are extra information. For centered inputs, only the last two terms remain. A first-order <uncertainty propagation> rule at zero input means would miss the product entirely; the GUM explicitly requires higher-order treatment when nonlinearity matters.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2]{ref}
An original counterexample keeps even the marginal shapes fixed. Let $Z$ have a standard <normal distribution> and let $S$ independently take values $+1$ and $-1$ with equal probabilities. Define $A=s_aZ$ and $T=s_\theta SZ$. Both have centered <normal distributions>, the quoted variances and zero <covariance>. They are not independent: their absolute standardized values are identical.
Their product is $s_as_\theta SZ^2$. Its <expected value> is zero, while its <variance> is $3s_a^2s_\theta^2$, because $\mathbb E[Z^4]=3$. Independent centered normal inputs with the same individual variances instead give product variance $s_a^2s_\theta^2$. The joint model, rather than a marginal histogram or correlation coefficient alone, determines the difference.
For an illustrative length correction $d=La\theta$ with zero means, $L=100\,\mathrm{mm}$, $s_a=10^{-6}\,\mathrm{K^{-1}}$ and $s_\theta=0.1\,\mathrm K$, these models give standard deviations $\sqrt3\times10\approx17.32$ nm and 10 nm. This constructed dependence is a mathematical test case, not a proposed universal thermometer model.
In a physical <gauge block> study, track paired temperature and coefficient information, shared thermometer corrections and any procedure that estimates one input using the other. NIST's thermal correction explicitly combines block length, expansion coefficient and temperature difference.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.4.3.1, Eq. 5.8]{ref} Preserve that dependence when generating joint draws. JCGM 101 distinguishes model formulation from propagation and gives a multivariate Gaussian input model as a specific choice, not a consequence of normal marginals alone.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.2 and 6.4.8]{ref} More simulation trials reduce numerical error conditional on the chosen model; they cannot identify unmeasured joint moments.
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