# Reducing the Stochastic Coal Output Model Using the Convolution of Probability Density Functions
## 4 Discussion  
Based on the research presented in Sections 2 and 3, calculations were made which the authors present in this section. We present the possibility of reducing stochastic models to create such models as an illustration of mining processes for the purpose of their further analysis in cases where the influence of geological-mining and technical-organizational conditions means that the selected process characteristics can be treated as random variables. A comprehensive description of such models and the methods for their solution are included, among others, in the works [22; 23]. The example (illustrated using general formulas) is taken from [3]. The daily output is the sum of each individual shift-wise output of production on the longwall face, i.e.,  
\[Q_{d}=\sum_{i=1}^{n}Q_{zi} \tag{18}\]  
where  
\(Q_{d}\)--the daily output [Mg/day];  
\(Q_{zi}\)--the shift-wise output on the \"i\" shift [Mg/m];  
\(n\)--the number of shift-wise outputs during the day [m/day]. Figure 14: Normal distribution as a function of \(f(t)\) variable \(T=T_{1}+T_{2}\). Source: own study. Figure 13: Examples of normal distributions as a function of \(f(t_{1})\) and \(f(t_{2})\) of variables \(T1\), \(T2\). Source: own study. According to the adopted assumptions, under certain conditions, the shift-wise output is treated as a random variable. The probability density of the sum of random variables is therefore the density of the marginal distribution of the variable \(Q_{d}\), which is a convolution of the random variable functions \(Q_{z_{i}}\), i.e.,  
\[f_{q_{d}}=f_{q_{z_{1}}}\bigotimes f_{q_{z_{2}}}\bigotimes\ldots\bigotimes f_{q_{ z_{n}}} \tag{19}\]  
where  
\(f_{q_{d}}\)--the function of the marginal distribution density of the random variable \(Q_{d}\) (daily output);  
\(f_{q_{z_{1}}}\)--the function of the probability density of the random variable \(Q_{z_{1}}\) (the shift-wise output on the first shift);  
\(f_{q_{z_{2}}}\)--the function of the probability density of the random variable \(Q_{z_{2}}\) (the shift-wise output on the second shift);  
\(f_{q_{z_{n}}}\)--the function of the probability density of the random variable \(Q_{z_{n}}\) (the shift-wise output on the shift \"\(n\)\"). According to the definition of the convolution of the probability density functions and taking into account the independence of the random variables \(Q_{z_{1}},\ Q_{z_{2}},\ \ldots,\ Q_{z_{n}}\), the general formula is obtained:  
\[f_{q_{d}}(q_{d})=\int\limits_{-\infty}^{\infty}\ldots\int\limits_{-\infty}^{ \infty}f_{q_{z_{1}}}(q_{z_{1}})f_{q_{z_{2}}}(q_{d}-q_{z_{1}})\ldots f_{q_{z_{n }}}(q_{d}-q_{z_{1}}-\cdots-q_{z_{n-1}})\ dq_{z_{1}}dq_{z_{1}}\ldots dq_{z_{n-1}} \tag{20}\]  
For example, for \(n=2\) (two shift-wise outputs from the longwall face from one day), the boundary function takes the form  
\[f_{q_{d}}(q_{d})=\int\limits_{-\infty}^{\infty}\ldots\int\limits_{-\infty}^{ \infty}f_{q_{z_{1}}}(q_{z_{1}})f_{q_{z_{2}}}(q_{d}-q_{z_{1}})dq_{z_{1}} \tag{21}\]  
the designations were adopted as shown in Formula (6). It can be seen that Formula (9) is identical to the formula marked as 2; hence, the stochastic model reduction described in this paper using the convolution of probability density function properties can be used to obtain the functional characteristics of the daily output from the longwall face. The conducted study confirms the effectiveness of using the convolution properties of probability density functions to reduce stochastic models in the context of mining production processes. The traditional stochastic simulations--although accurate--are often time-consuming and computationally intensive and may require extensive input data and calibration. The reduction method presented in this paper offers a promising alternative that retains mathematical rigor while significantly simplifying the analytical procedures. The model used in this research reflects a typical production scenario on a longwall face, where process segments--such as cutting, loading, and transport--are performed in sequential order.