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Unformatted text preview: Homework 7 36325/725 due Wednesday Oct 24 (1) Let . Find this to the bound you get from Chebyshev's inequality. (2) Let . Use Chebyshev's inequality to show that . (3) Let and . Bound using Chebyshev's inequality and using Hoeffding's inequality. Show that, when is large, the bound from Hoeffding's inequality is smaller than the bound from Chebyshev's inequality. (4) Let be iid with finite mean and finite variance . Let be the sample mean. The sample variance is defined to be (4a) Show that . (4b) Show that . Hint: Show that where and . Apply the large of large numbers to and to . Finally, use the facts about convergence of transformed random variables. (5) Let and only if be a sequence of random variables. Show that 1 e x g V v er r w3 e r g g pr @ ` $ 0
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for g j2ru er n H 3 p e #rl 3 d $ g pm oH h ( r g r r m H tl F1 w u r g (y g r g RC9r6 (y$XTx""UvctsU iqpqpp2ih a ( $ P W H b ( YXWVUSTSRQ fedcW p ( v Ird $q#ig T~ xq" r # } s` T~ Tq" r (r $ qpqpqpi e iG g r H r t t v H t 2ru l r e D ser n efhi( er n q D $ er n U(hd 5 g $ g e r iqpqpqpihh gg ( d k$jvihcfD ( " )'&%$#! . Compare (6) Let finite. Show that (7) Let Does (8) Hoeffding's inequality in action. Let (8a) Let be fixed and define Let Use Hoeffding's inequality to show that We call a confidence interval for . In practice, we truncate the interval so it does not go below 0 or above 1. (8b) Let's examine the properties of this confidence interval. Let and . Conduct a simulation study to see how often the interval contains (called the coverage). Do this for various values of between and . Plot the coverage versus . (8c) Plot the length of the interval versus . Suppose we want the length of the interval to be no more than . How large should be? p qH H y p y p H r y 3 p dEYx#r P 3 H @ (yS $ p( i 3 $ 8Urr y Urr y 9r r g r g VI9y p a x" H a 9r F (y$XTx"UvctsU ipqqppiG w u r g r r p e H Its `lP # e H 3 H tsf P ( r $ H r qpqppi e 2ih g 5 {i {Ut y 6 r r g ipqpqp2ih be iid and let . be a sequence of random variables such that converge in probability? Does converge in quadratic mean? . . Define 2 (G` QI5 g $ . Suppose that the variance is ...
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This note was uploaded on 05/25/2008 for the course STAT 36625 taught by Professor Larrywasserman during the Fall '01 term at Carnegie Mellon.
 Fall '01
 LarryWasserman

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