Jensen's Inequality

Theorem

Let

• $\varphi$ is convex on an open interval $I$
• $X$ : Random variable
• ${\mathcal{S}}_X$ : Support of $X$

If

• ${\mathcal{S}}_X\subseteq I$
• $E(X)$ is finite

Then $\varphi [E(X)]\le E[\varphi (X)]$