## 3 System Statistical Moments  
The performance function of the transposition system can be presented as function \(Z\), \(Z=g_{X}(\mathbf{X})\), and random variables can be depicted as \(\mathbf{X}=[X_{1}\), \(X_{2}\),..., \(X_{n}]^{T}\). In the transposition system, the performance function can be represented as \(Z=g_{X}(\mathbf{X})=\xi-E=\xi\) - \([\theta_{4}-(i_{12}i_{23}i_{34})\theta_{1}]\), in which \(\xi\) can be regarded as the threshold value. Random variables \(\mathbf{X}\) can be regarded as X- and Y-coordinates of gear 1, 2, 3, and 4 in Table 1. Function \(Z\) can be established by a surrogate model, such as the response surface method [37; 38]. Considering the computation and information quantity of function \(Z\), the Taylor series up to the third term is taken to truncate the performance function. Hence, \(g_{X}(\mathbf{X})\) at point \(\mathbf{x}^{*}\) can be indicated by  
\[Z=g_{X}(\mathbf{x}^{*})+(\mathbf{X}-\mathbf{x}^{*})^{T}\
abla g_{X}(x^{*})+\frac{1}{2}(\bm {X}-\mathbf{x}^{*})^{T}\
abla^{2}g_{X}(\mathbf{x}^{*})(\mathbf{X}-\mathbf{x}^{*})+o\Big{[}(\bm {X}-\mathbf{x}^{*})^{3}\Big{]} \tag{1}\]  
The last term \(o[\ ]\) is the higher-order infinitesimal remainder, which can be ignored. The second degree of the Taylor series is considered. The symbol \(\
abla g_{X}()\) with \(\mathbf{x}^{*}=\mathbf{\mu}\) can be defined by  
\[\
abla g(\mathbf{X})|_{X=\mathbf{x}^{*}=\mu}=\Big{[}\frac{\partial g(\mathbf{X})}{\partial X _{1}}\quad\frac{\partial g(\mathbf{X})}{\partial X_{2}}\quad\ldots\quad\frac{ \partial g(\mathbf{X})}{\partial X_{n}}\Big{]}_{X=\mu} \tag{2}\]  
Also, the symbol \(\
abla^{2}g_{X}()\) can be expressed as  
\[\
abla^{2}g(\mathbf{X})\Big{|}_{X=\mathbf{x}^{*}=\mu}= \begin{bmatrix}\frac{\partial^{ 2}g(\mathbf{X})}{\partial X_{1}^{2}}&\frac{\partial^{2}g(\mathbf{X})}{\partial X_{1} \partial X_{2}}&\ldots&\frac{\partial^{2}g(\mathbf{X})}{\partial X_{1}\partial X_{ n}}\\ \frac{\partial^{2}g(\mathbf{X})}{\partial X_{2}\partial X_{1}}&\frac{\partial^{2}g (\mathbf{X})}{\partial X_{2}^{2}}&\ldots&\frac{\partial^{2}g(\mathbf{X})}{\partial X_ {2}\partial X_{n}}\\ \frac{\partial^{2}g(\mathbf{X})}{\partial X_{n}\partial X_{1}}&\frac{\partial^{2}g (\mathbf{X})}{\partial X_{n}\partial X_{2}}&\ldots&\frac{\partial^{2}g(\mathbf{X})}{ \partial X_{n}^{2}}\end{bmatrix} _{X=\mu} \tag{3}\]  
The first four order center moments of \(\mathbf{X}\) can be depicted as \(\mu_{x1}\), \(\mu_{x2}\), \(\mu_{x3}\), and \(\mu_{x4}\) in turn. The \(\zeta^{th}\) center moment \(\mu_{x\zeta}\) can be expressed as \(\mu_{x\zeta}\)= \([\mu_{X_{1}}\zeta,\mu_{X_{2}}\zeta,...,\mu_{x\zeta}]\), \(\zeta\) = 1,2,3,4, and \(n\) = 1, 2, 3,.... Based on Equation (1), the mean value \(\mu_{z1}\) of performance function \(Z\) can be expressed as  
\[\mu_{Z1}=E[g(\mathbf{X})]=g(\mu_{x1})+\frac{1}{2}E\Big{[}(\mathbf{X}-\mu_{X1})^{T} \
abla^{2}g(\mu_{X1})(\mathbf{X}-\mu_{X1})\Big{]} \tag{4}\]  
in which  
\[ \begin{array}{lcl}E\Big{[}(\mathbf{X}-\mu_{X1})^{T}\
abla^{2}g(\mu_{X1})(\mathbf{X}- \mu_{X1})\Big{]}&=&\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}\frac{ \partial^{2}g(\mathbf{X})}{\partial X_{j}\partial X_{k}}\Big{|}_{X=\mu_{x1}}E\Big{[} \Big{(}X_{j}-\mu_{X1}\Big{)}\big{(}X_{k}-\mu_{X1}\big{)}\Big{]}\\ &=&\sum\limits_{j=1}^{n}\frac{\partial^{2}g(\mathbf{X})}{\partial X_{j}^{2}}\Big{|} _{X=\mu_{x1}}E\Big{(}X_{j}-\mu_{X1}\Big{)}^{2}=\sum\limits_{j=1}^{n}\mu_{Xj2} \frac{\partial^{2}g(\mathbf{X})}{\partial X_{j}^{2}}\Big{|}_{X=\mu_{x1}}\end{array}