A function g(x) is convex if the chord connecting any twopoints on the functions graph
Chapter 4, Problem 125(choose chapter or problem)
A function g(x) is convex if the chord connecting any twopoints on the functions graph lies above the graph.When g(x) is differentiable, an equivalent condition isthat for every x, the tangent line at x lies entirely on orbelow the graph. (See the figure below.) How doesg(m) 5 g(E(X)) compare to E(g(X))? [Hint: The equationof the tangent line at x 5 m is y 5 g(m) 1 g9(m) ? (x 2 m).Use the condition of convexity, substitute X for x, andtake expected values. [Note: Unless g(x) is linear, theresulting inequality (usually called Jensens inequality)is strict (, rather than # ); it is valid for both continuousand discrete rvs.]
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