Sep 21, 2017 term paper 2

PROVE MILL’S INEQUALITY, THEOREM 4.7. HINT. NOTE THAT P(|Z| > T) = 2P(Z>T). NOW WRITE OUT..

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5. Prove Mill’s inequality, Theorem 4.7. Hint. Note that P(|Z| > t) = 2P(Z>t). Now write out what P(Z>t) means and note that x/t > 1 whenever x>t. 6. Let Z ~ N(0, 1). Find P(|Z| > t) and plot this as a function of t. From Markov’s inequality, we have the bound P(|Z| > t) = E|Z| k tk for any k > 0. Plot these bounds for k = 1, 2, 3, 4, 5 and compare them to the true value of P(|Z| > t). Also, plot the bound from Mill’s inequality. 4.5 Exercises 69 7. Let X1,…,Xn ~ N(0, 1). Bound P(|Xn| > t) using Mill’s inequality, where Xn = n-1 n i=1 Xi. Compare to the Chebyshev bound


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