## 2 Materials and Methods
### Probabilistic Analysis  
In spite of the lack of literature regarding PQ event probabilities, it is clear that the research should be started with the Poisson distribution, which is usually used for event frequency assessment. The Poisson distribution is a discrete distribution whose probability mass function is:  
\[f(k,\lambda)=\frac{\lambda^{k}}{k!}\mathrm{exp}(-\lambda), \tag{9}\]  
where: \(\lambda\)--the expected value; \(k\)--the number of occurrences. On the other hand, since exact distribution types of specific event groups are often unknown, the alternatives should also be investigated. In the case of the Poisson distribution, the variance (of a random variable) is equal to \(\lambda\). However, PQ data sets can have various variances and distributions, but according to the central limit theorem, in many situations, these distributions must tend towards the normal. Therefore, in addition, let us investigate the Gaussian distribution whose probability density function is:  
\[f(x,\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\mathrm{exp}\left(-\frac{1}{2}\left( \frac{x-\mu}{\sigma}\right)^{2}\right), \tag{10}\]  
where: \(\mu\)--the mean; \(\sigma^{2}\)--the variance. The second step is the confidence interval calculation. According to the conventional approach, when sample size is not fairly large, say lower than 40 ([PERSON], 1954), the central limit theorem does not apply, thus Student's t-distribution must be used for the determination of the confidence interval:  
\[\left(\mu-t_{\alpha,\
u}\frac{s}{\sqrt{n}};\;\mu+t_{\alpha,\
u}\frac{s}{\sqrt {n}}\right), \tag{11}\]  
where: \(\mu\)--the mean; \(s\)--the sample's standard deviation; \(n\)--sample size; \(t_{\alpha,\
u}\)--the t-test critical value, which depends on the test type (either one-tailed or two-tailed), significance level \(\alpha\), and degrees of freedom \(\
u\) which are determined as follows:  
\[\
u=n-1. \tag{12}\]  
Standard deviation for a sample is calculated using the following formula:  
\[s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}, \tag{13}\]  
where: \(n\)--sample size; \(\overline{x}\)--sample mean. It should be noted that the following properties of both the expected value and the variance are important to follow during data aggregation:  
\[\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y], \tag{14}\]  
\[\text{Var}[X+Y]=\text{Var}[X]+\text{Var}[Y], \tag{15}\]  
where: \(X\) and \(Y\) are independent random variables. In addition to the conventional approach, a more specialized method for the Poisson distribution mean will also be used, when the confidence interval is expressed as follows:  
\[\frac{1}{2}\left(F_{\chi^{2}}^{-1}\Big{(}\frac{\alpha}{2},2k\Big{)};\;F_{\chi^ {2}}^{-1}\Big{(}1-\frac{\alpha}{2},2k+2\Big{)}\right), \tag{16}\]  
where: \(F_{\chi^{2}}^{-1}\)--the quantile function of chi-squared distribution; \(\alpha\)--the significance level (probability of a type I error); \(k\)--observation from the Poisson distribution with mean \(\mu\)[62]. The quantile function of the chi-squared distribution is:  
\[F_{\chi^{2}}^{-1}(x,\
u)=\frac{\Gamma\left(\frac{\
u}{2},\frac{1}{2k}\right)} {\Gamma\left(\frac{\
u}{2}\right)}=\frac{1}{\left(\frac{\
u}{2}-1\right)!} \int\limits_{\frac{1}{2k}}^{\infty}t^{\frac{\
u}{2}-1}\text{exp}(-t)dt, \tag{17}\]  
where: \(\
u\)--degrees of freedom; \(\Gamma\Big{(}\frac{\
u}{2},\frac{1}{2k}\Big{)}\)--the upper incomplete gamma function; \(\Gamma\Big{(}\frac{\
u}{2}\Big{)}\)--the gamma function. Since the chi-squared distribution is a special case of the gamma distribution, Equation (16) is equivalent to:  
\[\Big{(}F_{\Gamma}^{-1}\Big{(}\frac{\alpha}{2},k,\theta\Big{)};\;F_{\Gamma}^{-1 }\Big{(}1-\frac{\alpha}{2},k+1,\theta\Big{)}\Big{)}; \tag{18}\]  
where: \(F_{\Gamma}^{-1}\)--the quantile function of the gamma distribution; \(k\)--the shape parameter; \(\theta\)--the scale parameter equal to 1.