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Jensen's Inequality

Jensen's Inequality: the convex transformation of a mean is less than or equal to the mean after convex transformation.
if g is a convex function: g(E[X]) <= E[g(X)]

A graphical "proof" of Jensen's inequality for the probabilistic case. The dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly "stretches" the distribution for increasing values of X.