Let (h: boldsymbol{x} mapsto mathbb{R}) be a convex function and let (boldsymbol{X}) be a random variable. Use

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Let \(h: \boldsymbol{x} \mapsto \mathbb{R}\) be a convex function and let \(\boldsymbol{X}\) be a random variable. Use the subgradient definition of convexity to prove Jensen's inequality:

\[ \begin{equation*} \mathbb{E} h(\boldsymbol{X}) \geqslant h(\mathbb{E} \boldsymbol{X}) \tag{2.56} \end{equation*} \]

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