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 $[a_{ij}-\epsilon|a_{ij}|,a_{ij}+\epsilon|a_{ij}|]$.

$$ \widetilde{a}_{ij}=(1+\epsilon\xi_{ij})a_{ij} \tag{1} $$

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

$$ \begin{aligned} \min \quad & c^Tx\\ \textrm{s.t.} \quad & Ex=e\\ & Ax\leq b\\ & \sum_ja_{ij}x_j+\epsilon\left(\sum_{j\in J}|a_{ij}|y_{ij}+\Omega\sqrt{\sum_{j\in J_i}a_{ij}^2 z_{ij}^2}\right)\leq b_i+\delta\max[1,|b_i|]\\ & l\leq x \leq u\\ & -y_{ij}\leq x_{j} -z_{ij}\leq y_{ij} \end{aligned} $$

where $\Omega$ 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) ?

$$ \begin{aligned} &\mathrm{Pr}\left(\sum_j \tilde{a}_{ij}x_j> b_i+\delta\max[1,|b_i|]\right)\\ &=\mathrm{Pr}\left(\sum_j \tilde{a}_{ij}x_j> b_i^+\right)\\ &=\mathrm{Pr}\left(\sum_j a_{ij}x_j+\epsilon\sum_j \xi_{ij}|a_{ij}| x_i> b_i^+\right) \ \ (\because Eq. (1)　?)\\ &= \mathrm{Pr}\left(\sum_j a_{ij}x_j+\epsilon\sum_j \xi_{ij}|a_{ij}| (x_i-y_{ij})+\epsilon\sum_j \xi_{ij}|a_{ij}| y_{ij}> b_i^+\right)\\ &\leq \mathrm{Pr}\left(\sum_j a_{ij}x_j+\epsilon\sum_j \xi_{ij}|a_{ij}| z_{ij}+\epsilon\sum_j \xi_{ij}|a_{ij}| y_{ij}> b_i^+\right) \ \ (\because x_i-y_{ij}\leq z_{ij} ?) \\ &\leq \mathrm{Pr}\left(\sum_{j\in J_i}\xi_{ij}|a_{ij}|y_j>\Omega\sqrt{\sum_{j\in J_i}a_{ij}^2 y_{ij}^2}\right) (Eq. Q) \end{aligned} $$