In this case, the coefficient \(a\) of the linear function is determined by the \"least squares\" method using the MATHCAD 15 programming environment according to the methods of works [18; 19; 20] and according to the formula  
\[a=r_{yt\ell}\frac{\sigma_{y}}{\sigma_{\ell}}, \tag{7}\]  
where \(r_{yt\ell}\)--is the correlation coefficient between random numbers \(y\) and \(\ell\);  
\(\sigma_{y},\sigma_{\ell}\)--standard deviations of random numbers \(y\) and \(\ell\), respectively. The coefficient b for the regression equation is determined by the formula  
\[b=m_{y}-am_{\ell}, \tag{8}\]  
where \(m_{y}\)--is the average value of \(y\); \(m_{\ell}\)--is the average value of the operating time. The correlation coefficient \(r_{yt\ell}\) characterizes the tightness of the linear relationship between random variables \(y\) and \(\ell\), and is determined by the formula  
\[r_{yt\ell}=\frac{k_{y}\ell}{\sigma_{y}\sigma_{\ell}}=\frac{\alpha_{11}(y,\ell )-m_{y}m_{\ell}}{\sigma_{y}\sigma_{\ell}}, \tag{9}\]  
where \(\alpha_{11}(y,\ell)=\frac{1}{N}\sum_{i=1}^{N}y_{i}\ell_{i}\)--is the second mixed initial moment for random variables \(y\) and \(\ell\). For 3 VL80s electric locomotives operated by JSC Uzbekistan Temir Yollari (Tashkent, Uzbekistan) (according to the data of the Department of Operation of Locomotives for the Uzbekistan Depot), calculations were carried out on controlled parameters such as the thickness of the wheel pair tread and the thickness of the large-toothed wheel of the traction reducer (Figure 2). As a result of the numerical calculation of the coefficients of the empirical regression equations in the MATHCAD 15 programming environment using approximation and spline interpolation methods, a regression equation was obtained for the dependence on the mileage of the standard deviation of the wear of the tooth of the pinion of the large-toothed wheel of the traction reducer of the VL80s electric locomotive. \[\sigma_{x}(\ell)=0.0000171\mathrm{x}+0.2582241, \tag{10}\]  
where the correlation coefficient r = 0.195. We have conducted surveys of the traction transmission of VL-80s electric locomotives on emerging breakdowns and failures from 2017 to 2023, according to Uzbekistan depot data. The gearing of VL80-c electric locomotives is rigid, double-sided, helical, and consists of two gears and two pinions enclosed in pairs in a protective casing. Figure 3 shows the wheel-motor block (WMB) of a VL-80s electric locomotive with a double-sided helical gear with an all-roller gear. The advantages of helical gearing are the following:  
* The helical teeth, located at an angle of 24\({}^{\circ}\)37\({}^{\prime}\)12\({}^{\prime\prime}\), ensure simultaneous gearing of pinion and gear teeth on both sides (due to axial displacement of the TEE armature with two pinions in roller bearings by 6 \(\div\) 8 mm);
* The high smoothness of gearing, because the wheel teeth do not enter and exit the gear at once with their entire width b, but gradually;
* A reduction in noise and additional internal dynamic load. Figure 3: Wheel-motor block (WMB) of VL-80s electric locomotive with double-sided helical gear with one-piece rolled gear wheel. Figure 2: Distribution of the tooth thickness of the gear of the large gear wheel of the traction gearbox for the VL-80s electric locomotive (according to the locomotive repair depot Uzbekistan). The disadvantage of helical gears is the axial force tending to move the wheel with the shaft and the required axial fixation of the shaft. Therefore, the time angle is limited and is recommended to be taken as \(\beta=8\ldots 25^{\circ}\). According to the data of the works, the reject dimensions of the gear are as follows [4; 5; 6; 7]:  
1. Tooth thickness wear is allowed but not more than 3.5 mm, measured at a height of 10 mm from the top of the tooth;
2. Lateral (axial) clearance between the teeth of the gear and the gear wheel in gearing is allowed but not more than 5.5 mm;
3.