```markdown
### Example: Yield Function

**Equation:**
\[ Y(x) = \text{Prob}(x + w \in S) \]

**Definitions:**
- \( x \in \mathbb{R}^n \): nominal parameter values for product
- \( w \in \mathbb{R}^n \): random variations of parameters in manufactured product
- \( S \): set of acceptable values

**Properties:**
- If \( S \) is convex and \( w \) has a log-concave probability density function (pdf), then:
  - \( Y \) is log-concave
  - Yield regions \(\{ x \mid Y(x) \geq \alpha \}\) are convex
  - Not necessarily tractable though...

**Source:**
A. d'Aspremont. Convex Optimization M2.

**Slide:** 31/67
```