Jensen inequality

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Let LaTeX: f be convex on LaTeX: X, and let LaTeX: x be a random variable with expected value, LaTeX: E(x). Then, LaTeX: E(f(x)) \ge f(E(x)). For example, let LaTeX: X=R and LaTeX: f(x) = x^2. Suppose LaTeX: x is normally distributed with mean zero and standard deviation LaTeX: s. Then,

LaTeX: E(f(x)) = E(x^2) = s^2 \ge 0 = E(x) = f(E(x)).


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