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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 accompanying figure.) How does $g(\mu)=g[E(X)]$ compare to the expected value $E[g(X)] ?[$ Hint: The equation of the tangent line at $x=\mu$ is $y=g(\mu)+g^{\prime}(\mu) \cdot(x-\mu) .$ Use the condition of convexity, substitute $X$ for $x,$ and take expected values. Note: Unless $g(x)$ is linear, the resulting insually called Jensen's inequality is strict $(<$ rather than $\leq) ;$ it is valid for both continuous and discrete rvs.]

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

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