Explain why E(W) > 81 must hold by applying Jensens inequality
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Step 1: Understand Jensen's Inequality Jensen's Inequality states that for a convex function \( f \) and a random variable \( X \), the following holds: \[ E[f(X)] \geq f(E[X]) \] This means that the expected value of a convex function of a random variable is Show more…
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Ex. 1 — Let X be a positive random variable whose expected value is E[X] = 10 Find a lower bound to the probability P(X < 20) Ex. 2 — Let X be a random variable such that E[X] = 0, P(−3 < X < 2) = 1/2 Find a lower bound to its variance. Ex. 3 — Let X be a strictly positive random variable, such that E[X] = 1/2, Var[X] = 1 What can you infer, using Jensen’s inequality, about the following expected value: E[ln(2X)]
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