## 3 Results and Discussion
### Strength Analysis
The value of the Weibull modulus of the investigated sample is consistent with previously published studies [44; 45]. Typically, zirconia ceramics are reported to exhibit a Weibull modulus below 7, while that of alumina ceramics can reach up to 10 [24]. However, there are reports demonstrating that the \(m\) values of the zirconia, both ZrO\({}_{2}\)-Al\({}_{2}\)O\({}_{3}\) microparticle and nanoparticle composites, were in a range between 10.07 and 12.02 [46]. In general, a higher Weibull modulus value represents a narrower dispersion of the fracture strength, and thus a more homogeneous distribution of flaws, providing higher reliability [46]. Thus, a high modulus suggests that the nanopowder studied is high quality and reliable. The scale parameter \(\sigma_{0}\) equal to 461.15 MPa is indicative of characteristic strength, representing the stress level at which 63.2% of the population would fail. Figure 8 reveals that the studied sample exhibits a high breakdown strength of 461.15 MPa. The expected value to failure (mean value to failure) can be calculated based on the following equation:  
\[MVF=\sigma\Gamma\left(1+\frac{1}{m}\right) \tag{8}\]  
where G is a gamma function. The average durability of the sample in terms of tensile strength measured by the expected value to failure is equal to 432.91 MPa. The median and mode are 449.54 and 461.08 MPa, respectively, and the standard deviation does not exceed 37.86 MPa, Table 4.  
The mean and median values differ by 17 MPa. The median is a slightly more accurate estimator than the mean when the distribution is unsymmetrical. The reliability function, which represents the probability that the sample will keep its properties without failure, and the failure function, which represents the probability of failure, are shown in Figure 9 (Both were plotted based on the parameters from the linear fit shown in Figure 8). \begin{table}
\begin{tabular}{c c c c} \hline
**Mean, MVE, MPa** & **Median, MPa** & **Mode, MPa** & **Standard Deviation, MPa** \\ \hline
432.91 & 449.54 & 461.08 & 37.86 \\ \hline \end{tabular}
\end{table} 
Table 4: The statistical parameters of the Weibull distribution of the experimental bending strength for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{2}{c}{**Fitting Parameters**} & \multicolumn{3}{c}{**Statistics of the Fit**} & \multicolumn{3}{c}{**Weibull Distribution Parameters**} \\ \hline
**a\({}_{0}\)** & **a\({}_{1}\)** & **r** & **s** & **Reduced \(\chi^{2}\)** & **m** & \(\sigma_{0_{r}}\)**MPa** \\ \hline
14.383 \(\pm\) 7.4 \(\times\) 10\({}^{-6}\) & \(-\)88.222 \(\pm\) 1.36 \(\times\) 10\({}^{-5}\) & 0.999 & 0.028 & 1.99 \(\times\) 10\({}^{-10}\) & 14.383 \(\pm\) 7.4 \(\times\) 10\({}^{-6}\) & 461.15 \(\pm\) 3.11 \(\times\) 10\({}^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 3: The results of the parameter fitting for the best-fit formulation, along with the derived Weibull distribution parameters. The hazard function, which describes the \"conditional density\", i.e., the conditional distribution of a random variable, given that the event has not yet occurred, and the failure distribution, i.e., the probability density of failures, are shown in Figure 10.  
Figure 10: The hazard function (**top**) and probability density of failures (**bottom**) drawn based on the parameters from the linear fit shown in Table 3 for the experimental bending strength for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. The black points on the graph illustrate only the experimental data, the \(\sigma\) values correspond to the experimental and hazard function (**top**) or probability density of failures (**bottom**); values are calculated based on the Weibull model. Figure 9: The reliability and failure functions drawn based on the parameters from the linear fit shown in Table 3 for the experimental bending strength for ZrO\({}_{2}\)–5 wt.% CeO\({}_{2}\) sintered sample. The lines represent the model.