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X= X+ - where Z, is any random variable with E(Z) = en", where e>0 and a € R are fixed, and X is any other random variable. (a) Let e>0. Show that P(|Xn – X| > e) S P(|Z„] 2 en!/²). Hence, using Chebyshev's inequality, show that P(|Xn – X| > «) <ênl-a (b) For what values of a does the argument in part (a) prove that X converges in probability to X? (c) For the values of a identified in part (b), show that X, converges in mean square. (d) Why does your answer in part (c) imply the result in part (b)?

X= X+ - where Z, is any random variable with E(Z) = en", where e>0 and a € R are fixed, and X is any other random variable. (a) Let e>0. Show that P(|Xn – X| > e) S P(|Z„] 2 en!/²). Hence, using Chebyshev's inequality, show that P(|Xn – X| > «) <ênl-a (b) For what values of a does the argument in part (a) prove that X converges in probability to X? (c) For the values of a identified in part (b), show that X, converges in mean square. (d) Why does your answer in part (c) imply the result in part (b)?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Suppose that X, is a sequence of random variables such that Zm X, = X + n!/2' where Z, is any random variable with E(Z) = cn", where c> 0 and a € R are fixed, and X is any other random variable. (a) Let e > 0. Show that P(|X, – X| > e) < P((Z„|2 en!/²). Hence, using Chebyshev's inequality, show that P(\Xn – X| > e) < 2nl-a° (b) For what values of a does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of a identified in part (b), show that X„ converges to X in mean square. (d) Why does your answer in part (c) imply the result in part (b)?
Transcribed Image Text:Suppose that X, is a sequence of random variables such that Zm X, = X + n!/2' where Z, is any random variable with E(Z) = cn", where c> 0 and a € R are fixed, and X is any other random variable. (a) Let e > 0. Show that P(|X, – X| > e) < P((Z„|2 en!/²). Hence, using Chebyshev's inequality, show that P(\Xn – X| > e) < 2nl-a° (b) For what values of a does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of a identified in part (b), show that X„ converges to X in mean square. (d) Why does your answer in part (c) imply the result in part (b)?
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