```markdown
### Image Description

The image is a slide discussing a stochastic approach via Second Order Cone Programming (SOCP) in the context of optimization. The key points are:

- **Assumptions**: 
  - \( a_i \) is assumed to be Gaussian with mean \( \bar{a}_i \) and covariance \( \Sigma_i \) (i.e., \( a_i \sim \mathcal{N}(\bar{a}_i, \Sigma_i) \)).
  - \( a_i^T x \) is a Gaussian random variable with mean \( \bar{a}_i^T x \) and variance \( x^T \Sigma_i x \).

- **Probability Expression**:
  \[
  \text{Prob}(a_i^T x \leq b_i) = \Phi \left( \frac{b_i - \bar{a}_i^T x}{\|\Sigma_i^{1/2} x\|_2} \right)
  \]
  where \( \Phi(x) = (1/\sqrt{2\pi}) \int_{-\infty}^{x} e^{-t^2/2} dt \) is the cumulative distribution function (CDF) of \( \mathcal{N}(0,1) \).

- **Robust Linear Programming (LP)**:
  - Objective: Minimize \( c^T x \)
  - Subject to: \( \text{Prob}(a_i^T x \leq b_i) \geq \eta \), for \( i = 1, \ldots, m \).

- **Equivalence to SOCP**:
  - The robust LP problem is equivalent to the following SOCP:
  - Objective: Minimize \( c^T x \)
  - Subject to: \( \bar{a}_i^T x + \Phi^{-1}(\eta) \|\Sigma_i^{1/2} x\|_2 \leq b_i \), for \( i = 1, \ldots, m \).
  - Condition: \( \eta \geq 1/2 \).

- **Source**: A. d’Aspremont, Convex Optimization M2, Slide 59/67.
```
