125. A function g(x) is convex if the chord connecting any two points on the functions graph...
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125. A function g(x) is convex if the chord connecting any two points on the function’s graph lies above the graph. When g(x) is differentiable, an equivalent condition is that for every x, the tangent line at x lies entirely on or below the graph. (See the figures below.) How does g() g(E(X))
compare to E(g(X))? [Hint: The equation of the tangent line at x is y g() g () (x ). Use the condition of convexity, substitute X for x, and take expected values. Note:
Unless g(x) is linear, the resulting inequality (usually called Jensen’s inequality) is strict ( rather than ); it is valid for both continuous and discrete rv’s.]
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Related Book For
Probability And Statistics For Engineering And The Sciences
ISBN: 9781111802325
7th Edition
Authors: Dave Ellis, Jay L Devore
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