# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### The Probabilities of Correct and Incorrect Decisions at CBM Inspections
Integrating PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) within the limits, we obtain the probability of event \(\Lambda_{4}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{4}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n}}^{t_{n+1}}\int\limits_{ \infty}^{t_{n}}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{33}\]  
Event \(\Lambda_{5}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the following limits: \(0<\mathrm{H}\leq t_{n}\) and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n\). By integrating PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) within the specified region, we determine the probability of event \(\Lambda_{5}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{5}(t_{1},\ldots,t_{n})]=\int\limits_{0}^{t_{n}}\int\limits_{t_{n}}^ {\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1},\ldots, u_{n})du_{1}\ldots du_{n}d\vartheta \tag{34}\]  
Event \(\Lambda_{6}(t_{1},\ldots,t_{n})\) corresponds to the \((n+1)\)-dimensional region with the limits: \(0<\mathrm{H}\leq t_{n}\), \(-\infty<\mathrm{H}_{n}^{*}\leq t_{n}\), and \(t_{i}<\mathrm{H}_{i}^{*}<\infty\), \(i=1,\ldots,n-1\). Integrating PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) within the limits, we obtain the probability of event \(\Lambda_{6}(t_{1},\ldots,t_{n})\). \[P[\Lambda_{6}(t_{1},\ldots,t_{n})]=\int\limits_{0}^{t_{n}}\int\limits_{-\infty }^{t_{n}}\int\limits_{t_{n}-1}^{\infty}\ldots\int\limits_{t_{1}}^{\infty} \omega_{0}(\vartheta,u_{1},\ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{35}\]  
As we can observe from (30)-(35), the joint PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) of random variables \(\{\mathrm{H},\mathrm{H}_{1}^{*},\ldots,\mathrm{H}_{n}^{*}\}\) must be known in order to determine the probabilities of correct and incorrect decisions made when conducting CBM inspections. We denote the conditional PDF of random variables \(\Delta_{1},\ldots,\Delta_{n}\) as \(\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)\) provided that \(\mathrm{H}=\eta\). Following the multiplication theorem of PDFs, we represent PDF \(\omega_{0}\big{(}\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*}\big{)}\) as follows [94]:  
\[\omega_{0}(\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*})=\omega(\eta)\varphi(\eta_{1}^ {*},\ldots,\eta_{n}^{*}|\eta) \tag{36}\]  
where \(\varphi(\eta_{1},\ldots,\eta_{n}|\eta)\) is the conditional PDF of random variables \(\mathrm{H}_{1}^{*},\ldots,\mathrm{H}_{n}^{*}\), provided that \(\mathrm{H}=\eta\). In the case of \(\mathrm{H}=\eta\), the random variables \(\mathrm{H}_{1}^{*},\ldots,\mathrm{H}_{n}^{*}\) are defined as \(\mathrm{H}_{1}^{*}=\eta+\Delta_{1},\ldots,\)\(\mathrm{H}_{n}^{*}=\eta+\Delta_{n}\). The following equality is true because of the additive relationship between random variables \(\mathrm{H}\) and \(\Delta_{i}\)\((i=1,\ldots,n)\):  
\[\varphi(\eta_{1}^{*},\ldots,\eta_{n}^{*}|\eta)=\psi_{0}(\eta_{1}^{*}-\eta, \ldots,\eta_{n}^{*}-\eta|\eta) \tag{37}\]We obtain the following expression for the multidimensional PDF by substituting (37) into (36). \[\omega_{0}(\eta,\eta_{1}^{*},\ldots,\eta_{n}^{*})=\omega(\eta)\psi_{0}(\eta_{1}^{* }-\eta,\ldots,\eta_{n}^{*}-\eta|\eta\,) \tag{38}\]  
It is feasible to simplify (30)-(35) using (38). Inputting (38) into (30) gives the following.