How can we convert from Robust Linear Programming to SOCP?

Problem establishment

Suppose that in inequality constraint $i$ the true values ${\tilde{a}}_{ij},j\in J_i$, of uncertain data can range in the interval $[a_{ij}-ϵ|a_{ij}|,a_{ij}+ϵ|a_{ij}|]$.

$$\begin{array}{}\text{(1)}& {\tilde{a}}_{ij}=(1+ϵ{\xi }_{ij})a_{ij}\end{array}$$

Assume that $x$ can be extended to a feasible solution $(x,y,z)$ of the optimization problem

$$\begin{array}{rl}min& c^{T}x\\ \text{s.t.}& Ex=e\\ & Ax\le b\\ & \sum _j{a}_{ij}{x}_j+ϵ(\sum _{j\in J}|a_{ij}|y_{ij}+\mathrm{\Omega }\sqrt{\sum _{j\in J_i}{a}_{ij}^2{z}_{ij}^2})\le b_i+\delta max[1,|b_i|]\\ & l\le x\le u\\ & -y_{ij}\le x_j-z_{ij}\le y_{ij}\end{array}$$

where $\mathrm{\Omega }$ is a positibe parameter.

Question

I have been trying to derive the equation (Q), but I have failed.
How do we derive the equation (Q) ?

$$\begin{array}{rl}& \mathrm{Pr}(\sum _j{\tilde{a}}_{ij}{x}_j>b_i+\delta max[1,|b_i|])\\ & =\mathrm{Pr}(\sum _j{\tilde{a}}_{ij}{x}_j>b_i^{+})\\ & =\mathrm{Pr}(\sum _j{a}_{ij}{x}_j+ϵ\sum _j{\xi }_{ij}|a_{ij}|x_i>b_i^{+})(\because Eq.(1) ?)\\ & =\mathrm{Pr}(\sum _j{a}_{ij}{x}_j+ϵ\sum _j{\xi }_{ij}|a_{ij}|(x_i-y_{ij})+ϵ\sum _j{\xi }_{ij}|a_{ij}|y_{ij}>b_i^{+})\\ & \le \mathrm{Pr}(\sum _j{a}_{ij}{x}_j+ϵ\sum _j{\xi }_{ij}|a_{ij}|z_{ij}+ϵ\sum _j{\xi }_{ij}|a_{ij}|y_{ij}>b_i^{+})(\because x_i-y_{ij}\le z_{ij}?)\\ & \le \mathrm{Pr}(\sum _{j\in J_i}{\xi }_{ij}|a_{ij}|y_j>\mathrm{\Omega }\sqrt{\sum _{j\in J_i}{a}_{ij}^2{y}_{ij}^2})(Eq.Q)\end{array}$$