```markdown
### Jensen's Inequality

**Basic Inequality:** If \( f \) is convex, then for \( 0 \leq \theta \leq 1 \),

\[
f(\theta x + (1 - \theta) y) \leq \theta f(x) + (1 - \theta) f(y)
\]

**Extension:** If \( f \) is convex, then

\[
f(\mathbb{E} z) \leq \mathbb{E} f(z)
\]

for any random variable \( z \).

The basic inequality is a special case with discrete distribution:

\[
\text{Prob}(z = x) = \theta, \quad \text{Prob}(z = y) = 1 - \theta
\]

*A. d'Aspremont. Convex Optimization M2. 14/67*
```
