Proof of Jensen's inequality

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Jensen's inequality.  Let $X$ be a random variable.  Let $\phi$ be a convex function.  Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$
See List of Inequalities.

$\bullet$ Proof.

Let $l(x)=a+bx$ be the tangent line to the function, $\phi(x)$, at the point $\phi(\mathscr{E}X)$.  Since $\phi$ is a convex function, we have $\phi(x)\geq l(x)=a+bx$.  Then $$\mathscr{E}[\phi(X)]\geq\mathscr{E}(a+b\,X)= a+b\,\mathscr{E}X=\phi(\mathscr{E}X).$$

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