= Finite inverse moments in thickness models
\b[Finite inverse moments in thickness models] determine whether a positive but uncertain layer thickness permits a finite mean or <variance> after conversion to <density>. With fixed positive <mass> and <area>, write $\rho=K/\delta$, where $K=M/A$. A finite thickness mean does not ensure that $\operatorname E(\delta^{-1})$ or $\operatorname E(\delta^{-2})$ exists.
Suppose the thickness <probability density function> behaves near zero as $p(\delta)\sim C\delta^{a-1}$ with $C>0$. The small-thickness part of the inverse moment is proportional to
$$
\int_0^\varepsilon \delta^{-q}p(\delta)\,d\delta\sim C\int_0^\varepsilon\delta^{a-q-1}\,d\delta.
$$
For $q>0$, this converges exactly when $a>q$. Thus a finite mean <density> requires $a>1$, while finite <density> <variance> requires $a>2$, assuming the rest of the distribution is well behaved.
An exact example uses a <gamma distribution> with shape $a$ and scale $\theta$. Substitution into its density and the <gamma function> integral gives
$$
\operatorname E(\delta^{-q})=\theta^{-q}\frac{\Gamma(a-q)}{\Gamma(a)},\qquad a>q.
$$
The distribution and parameter conventions are specified by NIST; the inverse-moment calculation here follows directly from the integral.<NIST and SEMATECH e-Handbook of Statistical Methods>[§1.3.6.6.11, Gamma Distribution: density and gamma-function definition]{ref} For $a=2$ and $\theta=d/2$, mean thickness is $d$ and mean <density> is $2K/d$, yet its second moment diverges. A simulation can return finite sample variances without establishing a finite population <variance>.
A <lognormal distribution> provides a different positive-tail model. If $\log\delta\sim N(\mu,s^2)$, then $\operatorname E(\delta^{-q})=\exp(-q\mu+q^2s^2/2)$, finite for every finite positive $q$. NIST defines this logarithmic-normal parameterization and its <coefficient of variation>.<NIST and SEMATECH e-Handbook of Statistical Methods>[§1.3.6.6.9, Lognormal Distribution: parameterization and Common Statistics]{ref} With mean thickness $d$ and <coefficient of variation> $c$, mean <density> is $K(1+c^2)/d$. At $c=0.20$, it exceeds $K/d$ by 4%.
<Tail-model dependence of inferred density uncertainty> is therefore experimentally consequential. Choosing a convenient positive distribution changes the implied inverse moments. Justify near-zero behaviour from dimensional evidence or physical bounds rather than treating positivity or a stable finite simulation as sufficient validation.
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