## 2 Materials and Methods
### Derivation of the pdf Assigned to True Analyte Concentration in the Aliquot
\begin{split}\overline{\mathbf{r}}=\int_{\mathbf{0}}^{\mathbf{b}/(m+a)}& \frac{\mathbf{m}}{b}\cdot\mathbf{r}\cdot d\mathbf{r}\\ &+\int_{\mathbf{b}/(m+a)}^{b/m}\left(\frac{b(m+a)}{2\cdot a^{2}}\cdot \frac{\mathbf{1}}{\mathbf{r}^{2}}-\frac{\mathbf{b}^{2}}{3\cdot a^{2}}\cdot\frac{\mathbf{1}}{ \mathbf{r}^{2}}+\frac{6a^{2}\cdot m-(m+a)^{3}}{6a^{2}b}\right)\cdot\mathbf{r}\cdot d \mathbf{r}\\ &+\int_{\mathbf{b}/m}^{b/(m-a)}\left[\frac{b^{2}}{3\cdot a^{2}}\cdot \frac{\mathbf{1}}{\mathbf{r}^{2}}-\frac{b(m-a)}{2\cdot a^{2}}\cdot\frac{\mathbf{1}}{\mathbf{r} ^{2}}+\frac{(m-a)^{3}}{6a^{2}b}\right]\cdot\mathbf{r}\cdot d\mathbf{r}\end{split}  \tag{9}\]  
The mathematical expression of the expectancy of \(R\) is  
\[\overline{\mathbf{r}}=\frac{\mathbf{b}}{2\mathbf{a}}\left[\frac{m}{a}\cdot ln\left(\frac{ m^{2}-a^{2}}{m^{2}}\right)+ln\left(\frac{m+a}{m-a}\right)\right] \tag{10}\]  
where \(ln()\) is the natural logarithm. Considering that \(a/m\leq 10^{-2}\), then the expression of \(\overline{\mathbf{r}}\) in Equation (10) can be better perceived if its expression is expanded in Maclaurin series depending on the \(z\) = \(a/m\) variable, respectively:  
\[\overline{\mathbf{r}}\cong\frac{\mathbf{b}}{2\mathbf{a}}\left[\mathbf{z}+\frac{\mathbf{z}^{3}}{6} \right]=\frac{\mathbf{b}}{2m}\left[\mathbf{1}+\frac{\mathbf{a}^{2}}{6m^{2}}\right] \tag{11}\]  
Figure 1: Schematic drawing of the dependence of the upper limit of integration on the \(r\) value. The expression of the mean value of the analyte concentration in Equation (11) is similar to that of a uniform distribution, i.e., \(b/(2m)\), except for a tiny increase of less than \(1.7\times 10^{-5}\%\), depending mainly on the weighing uncertainty. The variance of the analyte concentration, denoted by \(\sigma^{2}\), is calculated as  
\[\sigma^{2}=\overline{r^{2}}-\overline{r}^{2} \tag{12}\]  
where  
\[\overline{r^{2}}=\int_{\mathbf{0}}^{\frac{b}{m-a}}\mathbf{r}^{2}\cdot f_{ \mathbf{R}}(\mathbf{r})\cdot d\mathbf{r}=\frac{b^{2}}{3\mathbf{a}^{2}}\cdot ln \left(\frac{m^{2}}{m^{2}-a^{2}}\right) \tag{13}\]  
The mathematical expression of the variance of the \(R\) variable is  
\[\sigma^{2}=\frac{b^{2}}{3\mathbf{a}^{2}}\cdot ln\left(\frac{m^{2}}{m^{2}-a^{2 }}\right)-\frac{b^{2}}{4\mathbf{a}^{2}}\left[\frac{m}{a}\cdot ln\left(\frac{m ^{2}-a^{2}}{m^{2}}\right)+ln\left(\frac{m+a}{m-a}\right)\right]^{2} \tag{14}\]  
Based on the same argument as in Equation (10), we used the Maclaurin series expansion of Equation (14) to obtain a more significant form of \(\sigma^{2}\), i.e.,  
\[\sigma^{2}\cong\frac{b^{2}}{\mathbf{12}a^{2}}\left(\mathbf{z}^{2}+\frac{ \mathbf{z}^{4}}{2}\right)=\frac{b^{2}}{\mathbf{12}m^{2}}\left(\mathbf{1}+ \frac{a^{2}}{2m^{2}}\right) \tag{15}\]  
The variance of \(R\) is close to the variance in cases where it is uniformly distributed, but it contains a supplementary positive term, which increases its value. The increasing rate is \(\frac{d^{2}}{2m^{2}}\cdot 100\%\), i.e., in practice, less than \(0.05\%\). In cases where \(\mathbf{z}\leq 10^{-2}\), the coefficient of variation (\(\boldsymbol{CV}\)) of \(R\) is  
\[\boldsymbol{CV}=\frac{\sqrt{\frac{1}{3}+\frac{a^{2}}{6m^{2}}}}{\left[1+\frac{ a^{2}}{6m^{2}}\right]}\cong\frac{\sqrt{3}}{3}\simeq\mathbf{0.577} \tag{16}\]  
The \(\boldsymbol{CV}\) value of \(R\) is specific to a uniform \(\boldsymbol{pdf}\) in cases where the weighing error is less than \(10^{-2}\) g. Otherwise, \(\boldsymbol{CV}\) will be a function of \(\mathbf{z=\boldsymbol{a}\boldsymbol{m}}\), whose derivative is positive; i.e., \(\boldsymbol{CV(z)}\) is the minimum for \(\mathbf{z=0}\) and increases as \(\mathbf{z}\) increases. The model described above can be applied in the case of an ideal concentration measurement where the laboratory provides an exact concentration value in the range [0, \(b/(m-a)\)]. Laboratory practice shows that any analytical procedure is characterized by a limit of detection (LOD), LOQ, an analytical range, and uncertainty of the result estimated as the standard deviation (SD) or expanded uncertainty [10, 32, 33].