## 3 System Statistical Moments
 \tag{5}\]  
Hence, the mean value \(\mu_{z1}\) of the transposition system can be presented by  
\[\mu_{Z1}=E[g(\mathbf{X})]=g(\mu_{X1})+\frac{1}{2}\sum\limits_{j=1}^{n}\mu_{Xj2} \left[\frac{\partial^{2}g(\mathbf{X})}{\partial X_{j}^{2}}\Bigg{|}_{X=\mu_{x1}}\right] \tag{6}\]  
\[\mu_{x\pm}=\mathrm{E}\Big{\{}[\mathbf{X}-\mathrm{E}(\mathbf{X})]^{[4]}\Big{\}}= \epsilon^{4}\Big{[}\mathbf{X}_{p}^{[4]}\Big{]} \tag{12}\]  
in which \((\cdot)^{[6]}=(\cdot)^{[\beta-1]}\otimes(\cdot)=(\cdot)\otimes(\cdot)\otimes \ldots\otimes(\cdot)\) is the Kronecker operator, and \(\otimes\) is regarded as the Kronecker product, \((\boldsymbol{A})_{p\times q}\otimes(\boldsymbol{B})_{s\times t}=[a_{ij} \boldsymbol{B}]_{p\times ql}\). The relation, from Equation (10) to Equation (12), can be utilized to simplify the equations of high statistical moments in Section 3. Neglecting the terms \(\epsilon^{7}\) with \(\gamma\geq 3\), the covariance \(\mu_{z2}\), Equation (7), of the performance function \(Z\) can be expressed as  
\[\mu_{Z2}=\sum_{i=1}^{n}\Bigg{[}\left.\frac{\partial g(\mathbf{X})}{\partial X_ {i}}\right|_{X=\mu_{z1}}\Bigg{]}^{2}\mu_{X_{i}2}+o\Big{(}\epsilon^{3}\mathbf{X }_{p}^{[3]}\Big{)}+o\Big{(}\epsilon^{4}\mathbf{X}_{p}^{[4]}\Big{)}\approx\sum _{i=1}^{n}\Bigg{[}\left.\frac{\partial g(\mathbf{X})}{\partial X_{i}}\right|_ {X=\mu_{z1}}\Bigg{]}^{2}\mu_{X_{i}2} \tag{13}\]  
Also, the third center moments of the function \(Z\) can be expressed as  
\[\mu_{Z3}=\sum_{i=1}^{n}\Bigg{[}\left.\frac{\partial g(\mathbf{X})}{\partial X _{i}}\right|_{X=\mu_{z1}}\Bigg{]}^{3}\mu_{X_{i}3}+o\Big{(}\epsilon^{4}\mathbf{X }_{p}^{[4]}\Big{)}\approx\sum_{i=1}^{n}\Bigg{[}\left.\frac{\partial g(\mathbf{ X})}{\partial X_{i}}\right|_{X=\mu_{z1}}\Bigg{]}^{3}\mu_{X_{i}3} \tag{14}\]  
Considering the relation between \(\mu_{X_{i}2}^{2}\) and \(\mu_{X_{i}2}\mu_{X_{i}2}\) in Equation (9), partial items can be offset by each other. Hence, the fourth moment \(\mu_{z4}\) of the function \(Z\) can be approximated as  
\[\mu_{Z4}=\mathrm{E}\Big{[}(Z-\mu_{Z1})^{4}\Big{]}\approx\sum_{i=1}^{n}\Bigg{[} \left.\frac{\partial g(\mathbf{X})}{\partial X_{i}}\right|_{X=\mu_{z1}} \Bigg{]}^{4}\mu_{X_{i}4} \tag{15}\]  
Hence, \(\mu_{z2}\), \(\mu_{z3}\), and \(\mu_{z4}\) can be simplified by Equations (13), (14), and (15) approximately and conveniently. The statistical moment \(\mu_{z1}\) can be calculated by Equation (6). Those equations can be utilized to evaluate the statistical characteristics of the system's response. Considering the arbitrary distribution of system response \(Z\), the Edgeworth series is considered to represent the cumulative distribution function \(F(Z)\) as follows:  
\[F(Z)=\Phi(Z)-\varphi(Z)\Bigg{[}\frac{1}{8}\frac{\mathrm{E}\alpha_{H}}{\mu_{Z 2}^{2}}H_{2}(Z)+\frac{1}{24}\bigg{(}\frac{\mu_{Z4}}{\mu_{Z2}^{2}}-3\bigg{)}H_{ 3}(Z)+\frac{1}{22}\bigg{(}\frac{\mu_{Z2}}{\mu_{Z2}}\bigg{)}^{2}H_{5}(Z)+\cdots \Bigg{]} \tag{16}\]  
where functions \(\Phi()\) and \(\varphi(0)\) are the cumulative distribution function (CDF) and probability density function (PDF) of the normal distribution, respectively. Function \(H_{j}(\mathrm{Z})\) is the \(j^{\mathrm{th}}\) Hermite polynomial function. The recursion formula can be calculated using  
\[\left\{ \begin{array}{l}H_{j+1}(Z)=ZH_{j}(Z)-jH_{j-1}(Z)\\ H_{0}(Z)=1,\ H_{1}(Z)=Z\end{array} \right. \tag{17}\]  
Hence, the reliability index can be expressed as \(R(\beta)=[g(\mathbf{X})>0]=1-F(-\beta)\), and \(\beta=\mu_{z1}(\mu_{z2})^{-1/2}\). The reliability index \(R(\beta)\) can be utilized to analyze the safety of system performance in engineering. According to the simplified statistical moments, the CDF of the system responses can be used for the reliability evaluation of the transposition system.