# Reducing the Stochastic Coal Output Model Using the Convolution of Probability Density Functions
## 3 Results  
This section presents a step-by-step methodology for reducing the stochastic model for coal output using the convolution of probability density functions.  
The definition of the convolution of probability density functions is as follows [22]:  
**Definition 1.**_If \(X\) is an independent random variable with a distribution \(f_{x}(x)\), and \(Y\) is an independent random variable with a distribution \(f_{y}(y)\), then the random variable \(U\) is their sum, that is,_  
\[U=X+y \tag{13}\]  
_has a probability distribution_  
\[f(u)=\int\limits_{-\infty}^{\infty}f_{y}(y)f_{x}(u-y)dy \tag{14}\]  
Figure 6: Example of normal distribution function. Source: own study based on [25].  
The integral on the right-hand side of the formula is called the convolution integral. We can therefore say that the density of the sum of independent random variables is the convolution of the density of the summed random variables, which can also be written as  
\[f_{x+y}=f_{x}\bigotimes f_{y} \tag{15}\]  
The formula can be generalized to the case of summing any finite number of continuous random variables \(W_{1}\), \(W_{2}\ldots,W_{n}\) obtaining  
\[f_{W_{1}+W_{2}+\ldots+W_{n}}=f_{W_{1}}\bigotimes f_{W_{2}}\bigotimes\ldots \bigotimes f_{W_{n}} \tag{16}\]  
The following section presents a simple example of a stochastic model described using continuous random variables, which can be analyzed, for example, through stochastic simulation, or the model can be simplified using the properties of the convolution of probability density functions.  
Figure 7 shows an example fragment of a stochastic model, in which the random variables describing it are continuous.  
The model presents two serially performed operations, e.g., technological ones. The first operation is performed between the points marked with numbers 1 and 2, with the second occurring between 2 and 3. Each of these operations is characterized by a random variable, marked as \(T_{1}\) and \(T_{2}\). Their probability density functions have the symbols \(f(t_{1})\) and \(f(t_{2})\), respectively.  
Let us assume that the variables T1 and T2 characterize the time of execution of serially implemented technological operations (for some reasons, this time may therefore change).  
The characteristic \(f(t)\) of the random variable \(T\) is sought, where  
\[T=T_{1}+T_{2} \tag{17}\]  
The variable \(T\) is--in this case--the total time of the execution of a fragment of the process, e.g., production, in which two technological operations are carried out in series between points 1 and 3.  
A sample algorithm for obtaining the characteristic \(f(t)\) of the random variable T by means of stochastic simulation is presented in Figure 8.  
A fragment of the process model presented in Figure 7 can be reduced without the need to use the stochastic simulation algorithm described in Figure 8.  
The following part presents cases of model reduction using the properties of convolutions of specific types of continuous distributions (the examples concern the chi-square, gamma, and normal distributions).  
Figure 7: An example fragment of a stochastic model. Source: own study.  
The rules for selecting \"\(k\)\" were determined in accordance with the principles of statistics. A detailed description is provided in [22].