Proofs
Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered. Before embarking on these mathematical derivations, however, it is worth analyzing an intuitive graphical argument based on the probabilistic case where X is a real number (see figure). Assuming a hypothetical distribution of X values, one can immediately identify the position of and its image in the graph. Noticing that for convex mappings the corresponding distribution of Y values is increasingly "stretched out" for increasing values of X, it is easy to see that the distribution of Y is broader in the interval corresponding to X > X0 and narrower in X < X0 for any X0; in particular, this is also true for . Consequently, in this picture the expectation of Y will always shift upwards with respect to the position of, and this "proves" the inequality, i.e.
with equality when φ(X) is not strictly convex, e.g. when it is a straight line, or when X follows a degenerate distribution (i.e. is a constant).
The proofs below formalize this intuitive notion.
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