## 3 Results and Discussions
### Practical Application
These functions take values in the range [0, 1] because when \(t\rightarrow\infty\), \(R(t)=0\), \(F(t)=1\) and when \(t\to 0\), \(R(0)=1\), \(F(0)=0,f(t)\) verifies the relationship \(\int_{0}^{\infty}f(t)dt=1\). The dental burs' reliability can also be specified with the help of the numerical values of the indicators attached to the operating time, \(t\) (variable), until the first failure. These indicators (presented above) are \(M(t)\), \(D(t)\), \(\sigma\), and \(t_{\alpha}\), etc. The values, \(M(t)\), of the operation time without failure are determined with relation (4) and presented in Table 2. By integration of the function, \(M(t)\), \(t=20,433\) h was obtained in the working conditions at the rotation speed of 7000 rpm. When the system (the dental bur active part) is without reconditioning, then \(M(t)\) represents the average time without system failure, until the first failure. \(M(t)\) of the system, i.e., the average value of \(t\), can be expressed either by function \(R(t)\) or function \(f(t)\) (see relation (4)). The variation in the random variable, \(t\), or the distribution variation, \(D^{2}(t)\), is defined by relation (5) and is obtained through \(D^{2}(t)=2.562\) h, and the mean square deviation, \(\sigma\), of \(t\) is the indicator that can often be used instead of \(D^{2}(t)\) and results from \(\sigma=\sqrt{D^{2}(t)}=1.600\) h.  
\(D^{2}(t)\) and \(\sigma\) show the degree of uniformity of system/product performance (here, dental burs of the same type) from the point of view of reliability, by their values. If the  
Figure 11: Plots of reliability functions, (\(R(t)\) and failure, \(F(t)\), for the tested dental bur active parts. Figure 12: Frequency function plot, \(f(t)\), for the tested dental burs. technological process of manufacturing the system/product is well established, then the \(D^{2}(t)\) and \(\sigma\) values are relatively small and show us the degree of uniformity of the set of tested dental burs, as in the present case. The last important reliability indicator considered is the operating time quartile, \(t_{a}\), defined as the root of the equation:  
\[F(t_{a})=\alpha, \tag{12}\]  
and what results from equation/relation (6) and \(\alpha\) with values from 0.01, \(\ldots\), 0.99 is defined as a percentile. At a rotation speed of 7000 rpm and \(\alpha\) = 0.99, \(t_{a}\) = 21,502 h was obtained. In the reliability theory, the value his this \(t_{a}\) is interpreted as a guaranteed time, i.e., the period when the defective elements proportion in a certain system/group does not exceed the value, \(\alpha\) pre-set initially established [4]. The value of \(t_{a}\) = 21,502 h is very close to \(t_{2}\) = 21,573 h, calculated analytically, which justifies the correlation between the theoretical and experimental results. As stated above, the first data collected regarding the values of the investigated parameter (the lost mass by the dental bur active part) is presented as a disordered mass, and by ordering (ascending) of the experimental data, the statistical series is obtained. The grouping of the data is conducted on grouping intervals, whose number, \(n\), calculated with relation (7) is \(n\) = 5 (it must be an integer and was considered to be \(Z\) = 16 tested dental burs) and whose size is generally taken to be equal, except for the extreme intervals, when it can use different values. The size (amplitude) of the interval is determined based on relation (8) and is \(A\) = \((1.4-0.5)/5\) = 0.18. Another important role in the reliability theory also has the function, \(h(t)\) = \(f(t)\)/\(R(t)\), \(R(t)\) \(\
eq\) 0, that measures the intensity (rate) of failure (the instantaneous risk of failure), or the hazard of a system/component product, defined as the failure rate limit in the interval [\(t\), \(t+\Delta t\)], \(\Delta t\) \(>\) 0 being very small [4], whose variation is presented in Figure 13, which was used. It can be seen that the failure risk rate is around 10%, for 20 h, and increases approximately exponentially in the 21 st hour of operation.