## 3 Results and Discussions  
### Theoretical Aspects  
The distribution law establishes the correspondence between the random values, \(x_{i}\), and the respective probabilities, \(p_{i}\), using an analytical relationship as follows: \(P(X=x_{i})=P(x_{i})=p_{i}\) called the probability of failure or failure function, \(F(x)\), which can be empirical or theoretical. If the phenomenon/process probability, \(P(X=x_{i})\), is obtained after the experience based on the statistical observation data, then the statistical probability is a relative frequency, and the distribution is empirical. The failure function, \(F(x)\), is the most representative characteristic of a random variable and is defined as the phenomenon/process probability, so that the random variable, \(X<x\), is given/known:  
\[F(x)=P(X<x)=\sum_{i=1}^{x_{i}\leq x}p_{i} \tag{1}\]  
If \(X\) follows a continuous statistical distribution, \(F(x)\) has the form as follows:  
\[F(x)=P(X<x)=\int_{0}^{+\infty}f(x)dx=\int_{0}^{+\infty}Ft(x)dx=1-R(x) \tag{2}\]  
where \(f(x)\) is the frequency function (probability distribution density), which is the derivative of the function, \(F(x)\), and \(R(x)\) is the reliability function. The plot of the function, \(f(x)\), is illustrated in Figure 3.  
Figure 3: Variation plot of the function, \(f(x)\), for continuous functions. The size value, \(f(x)dx\), represents the probability that the variable, \(x\), belongs to the elementary interval, \(dx\). The function plot, \(F(x)\), from Figure 4 looks like the following:  
\[0<F(x)<1\text{ and }F(\infty)=\int_{0}^{+\infty}f(x)dx=1, \tag{3}\]  
that is, the area bounded by the function curve, \(f(x)\), and the abscissa axis are equal to unity. When studying a quality characteristic, the experimenter has at their disposal observation data obtained by measurements of durability. This durability is a random variable. Even if its distribution is not known, certain indicators and certain measures [7] of the trend of the considered variable are necessary. An experimental distribution (obtained through observations) that is, respectively, empirical shows a variation tendency with two aspects for any characteristic of the population: localization around a certain value (position) and, respectively, variation (scattering), as shown in Figure 5.  
It is noted that the distribution can be symmetrical or asymmetrical about the location area. The statistical indicators with which a quantitative analysis can be performed to compare trends in location and variation (dispersion) are as follows:  
1. Location indicators--these indicate the value where the real data of the phenomenon/ process tends to cluster, and the main indicator is the average [25] (with arithmetic average being the most important), \(M(X)\), and \[M(X)=\int_{0}^{\infty}[1-F(x)]dx\quad=\int_{0}^{\infty}R(x)dx=\quad\int_{0}^{ \infty}xf(x)\quad dx,\] (4)
2. Variation indicators--these represent the deviation of the values, \(x\), from their arithmetic mean and include the following:  
Figure 4: Distribution function continuous, \(F(x)\). Figure 5: Representation of distribution functions for phenomena/processes with different scatters. * Dispersion, \(D(X)\) or \(\sigma^{2}\) (\(\sigma=\sqrt{D(X)}\))--this represents the square mean deviation, and if it is defined in ratio with the mean value, \(\overline{x}\), the mathematical relationship is as follows: \[D^{2}(X)=\int_{0}^{\infty}\left(x-M(x)\right)^{2}\!f(x)dt.\] (5) The average squared deviation, \(\sigma\), calculated is a \"guarantee\" of the accuracy of the determinations only if there are at least three values [8]. The calculation is made only if the distribution is normal (Gaussian) or almost Gaussian. In addition, the value of \(\sigma\) allows conclusions to be drawn only if these come from at least 30 values simultaneously.