# Classification of Systems and Maintenance Models
## 4 Classification of Maintenance Models
### The Probabilities of Correct and Incorrect Decisions at CBM Inspections
\[P[\Lambda_{1}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{\infty}\omega(\vartheta )\int\limits_{t_{n}}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\psi_{0}(u_{1} -\vartheta,\ldots,u_{n}-\vartheta|\vartheta)du_{1}\ldots du_{n}d\vartheta \tag{39}\]  
Considering that \(\xi_{i}=u_{i}-\vartheta\) in (39), we arrive at:  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]0  
By changing the variables in (31) to (35), we obtain:  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]1  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]2  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]3  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]4  
\[P[\Lambda_{2}(t_{1},\ldots,t_{n})]=\int\limits_{t_{n+1}}^{t_{n}}\int\limits_{ \infty}^{\infty}\ldots\int\limits_{t_{1}}^{\infty}\omega_{0}(\vartheta,u_{1}, \ldots,u_{n})du_{1}\ldots du_{n}d\vartheta \tag{31}\]5  
As can be seen from (40) to (45), it is necessary to know the PDF \(\omega(\eta)\) and \(\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)\) to calculate the probabilities of correct and incorrect decisions made when conducting CBM inspections. Another thing to keep in mind is that formulas (40)-(45) are generalized, meaning that they may be applied to any stochastic deterioration process \(X(t)\). For brevity, we introduce the following notation for the probabilities \(P[\Lambda_{1}(t_{1},\ldots,t_{n})],\ldots,\)\(P[\Lambda_{6}(t_{1},\ldots,t_{n})]\):  
\[\left\{ \begin{array}{l}P_{TP}(t_{1},\ldots,t_{n})=P[\Lambda_{1}(t_{1},\ldots,t _{n})],\ P_{FN}(t_{1},\ldots,t_{n})=P[\Lambda_{2}(t_{1},\ldots,t_{n})]\\ P_{FP,1}(t_{1},\ldots,t_{n})=P[\Lambda_{3}(t_{1},\ldots,t_{n})],P_{TN,1}(t_{1}, \ldots,t_{n})=P[\Lambda_{4}(t_{1},\ldots,t_{n})]\\ P_{FP,2}(t_{1},\ldots,t_{n})=P[\Lambda_{5}(t_{1},\ldots,t_{n})]\,,\ P_{TN,2}(t_{1}, \ldots,t_{n})=P[\Lambda_{6}(t_{1},\ldots,t_{n})]\end{array} \right. \tag{46}\]  
where TP, FN, FP, and TN represent the true positive, false negative, false positive, and true negative, respectively. We use the time axis depicted in Figure 3 to derive the conditional probabilities of correct and incorrect decisions. When performing a CBM inspection at time \(t_{n}\)\((n=1,\ldots,k-1)\), the conditional probability of a \"true positive\" is defined as follows. \[P_{TP}(t_{1},\ldots,t_{n}|\eta\,)=P\left(\int\limits_{i=1}^{n}\Pi_{i}^{*}>t_{i} |\mathrm{H}=\eta\,\right) \tag{47}\]When running a CBM inspection at time \(t_{n}\) (\(n=1,\ldots,k-1\)), the conditional probability of a \"false negative\" is defined as follows. \[P_{FN}(t_{1},\ldots,t_{n}|\eta)=P\binom{n-1}{i=1}\mathrm{H}_{i}^{*}>t_{i}\cap \mathrm{H}_{n}^{*}\leq t_{n}|\mathrm{H}=\eta\ \bigg{)} \tag{48}\]  
When doing a CBM inspection at time \(t_{k}\) (\(k=1,\ldots,N\)), the conditional probability of a \"false positive 1\" event is defined as follows. \[P_{FP,1}(t_{1},\ldots,t_{k}|\eta)=P\binom{k}{n=1}\mathrm{H}_{n}^{*}>t_{n}| \mathrm{H}=\eta\ \bigg{)} \tag{49}\]  
When conducting a CBM inspection at time \(t_{k}\) (\(k=1,\ldots,N\)), the conditional probability of a \"true negative 1\" occurrence is described as follows.