Jensen's inequality

Let be a random variable and let be a convex function. Then, the expected value of satisfies the following inequality, called Jensen's inequality:

The weak inequality becomes a strict inequality If the function is strictly convex and is not a constant with probability one. When the function is concave (or strictly concave), then the weak (strict) inequality is reversed.

More details about Jensen's inequality - as well as a proof of the inequality and some exercises - can be found in the lecture entitled Basic probabilistic inequalities.

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