Theorem (Jensen’s Inequality)

Let $K\subset\mathbb{R}$ (interval) and let $f:K\to\mathbb{R}^n$ be convex function. Also let $X$ be a random vector with alphabet $\mathcal{X}^n\subset K$ and finite component means, then $$E[f(X)]\ge f(E[X])$$ Also, if $f$ is strictly convex then the inequality is strict unless $X$ is deterministic.