To check the compliance of a sample with the assumed theoretical distribution law, one can use one of the known criteria, for example, the Pearson goodness-of-fit criterion. It allows one to determine the probability that, due to random causes, the measure of divergence between the theoretical and statistical distributions will be greater than the actually observed one [19]. The samples of the controlled parameters of wear parts correspond to the normal distribution law. The distribution density for this law is as follows:  
\[f(x)=\frac{1}{\sigma_{x}\cdot\sqrt{2\pi}}\cdot e^{-\frac{(x-\sigma_{x})^{2}} {2\sigma_{x}^{2}}}, \tag{1}\]  
where \(m_{x}\) is the mathematical expectation of the value of the controlled parameter; \(\sigma_{x}\)--is the standard deviation of the value of the controlled parameter; \(x\)--is the current value of the controlled parameter. With the normal distribution law, the following relationships are imposed on a random variable:  
\(\sum_{j=1}^{k}P_{j}^{*}=1\)--the sum of frequencies over all intervals is equal to 1 (k is the number of intervals); \(m_{x}=m_{x}^{*}\); \(D_{x}=D_{x}^{*}\)--parameters of statistical and theoretical distributions. Since the numerical characteristics of the normal distribution law can be expressed through the mathematical expectation and variance of a random variable, the corresponding estimates are calculated: the average value \(m_{x}^{*}\) is equal to  
\[m_{x}^{*}=\frac{1}{N}\cdot\sum_{i=1}^{N}x_{i} \tag{2}\]  
and the estimate of the dispersion (standard deviation \(\sigma_{x}^{*}\)) is  
\[\sigma_{x}^{*}=\sqrt{\frac{1}{N-1}\cdot\sum_{i=1}^{N}((x_{i}-m_{x}^{*}))^{2}}, \tag{3}\]  
where N is the sample size of the controlled parameter; \(x_{i}\)--is the value of the controlled parameter. The distribution density functions of the tooth thickness of the large gear wheel of the traction reducer with fixed operating times of the 3 VL-80c series electric locomotive in the locomotive repair depot of Uzbekistan are presented in Figure 1. The values of the numerical characteristics of the distribution law of the controlled parameter make it possible to predict their changes at the greatest operating times, which in turn determines the resource, and for this purpose, analytical dependencies of the average values \(m_{x}\) and standard deviations \(\sigma_{x}\) on the mileage are found. The analytical dependence can be represented as a certain nonlinear function \(y=f(a_{1},a_{2},\ldots,a_{S},\ell_{i})\) of one argument \(\ell_{i}\), the expression of which includes \(S\) parameters \(a_{1},a_{2},\ldots,a_{S}\). Using this function, it is necessary to approximate the empirical regression specified in the form of n points \((\ell_{i},y_{i})\) for \(i=1,2,\ldots,n\), where y is understood as one of the parameters of the distribution law under consideration. The parameters of the function y are found by the least squares method, the condition of which is written as  
\[Z(a_{1},a_{2},\ldots,a_{S})=\sum_{i=1}^{n}\left[f(a_{1},a_{2},\ldots,a_{S}, \ell_{i})-y_{i}\right]^{2}=>min \tag{4}\]The values of the controlled parameters and the empirical dependencies \(m_{x}^{*}(\ell)\) and \(\sigma_{x}^{*}(\ell)\) of the wearing parts of traction rolling stock obtained in practice can be described by linear functions [2; 3; 4; 9; 10; 11; 18; 19; 20], that is, the approximating function has the form  
\[y=a\ell+b. \tag{5}\]  
The criterion for the conformity of the approximating function to the empirical dependencies is the minimum sum of the squares of the deviations of the empirical and theoretical functions  
\[\sum_{i=1}^{n}\left([y_{i}-(a\ell_{i}+b)]^{2}\right)\to min\,. \tag{6}\]  
Here  
\(y_{i}= \begin{cases}m_{xi}\\ \sigma_{xi}\end{cases} \) when approximating dependencies \( \begin{cases}m(\ell)\\ \sigma(\ell)\end{cases} \);  
\(\ell_{i}\)--operating time.