This exercise will lead you through a proof of Chebyshev's

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

ISBN: 9780073401331 38

Solution for problem 33SE Chapter 2

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 33SE

This exercise will lead you through a proof of Chebyshev's inequality. Let X be a continuous random variable with probability density function f(x). Suppose that P(X<0) = 0, so f(x) = 0 for x ≤ 0.

a. Show that

b. Let k> 0 be a constant. Show that

${\mu{}}_{x}\geq{}\int\limits_{k}^{\infty{}}kf(x)dx=kP(X\geq{}k)$

c. Use part (b) to show that P(X ≥ k) ≤ μX / k. This is called Markov's inequality. It is true for discrete as well as for continuous random variables.

d. Let Y be any random variable with mean μY and variance Let X = (Y − μY)2. Show that

e. Let k > 0 be a constant. Show that

f. Use part (e) along with Markov's inequality to prove Chebyshev's inequality: P(|Y − μY| ≥ kσY) ≤ 1/k2.

Step-by-Step Solution:

Answer

Step 1 of 6

a) Let x is a continuous random variable with probability density function f(x)

Here we have to show that

E(X)

Hence proven that

Step 2 of 6

Chapter 2, Problem 33SE is Solved

Step 3 of 6

Textbook: Statistics for Engineers and Scientists

Edition: 4

Author: William Navidi

ISBN: 9780073401331

This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. This full solution covers the following key subjects: show, let, inequality, random, Continuous. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. The full step-by-step solution to problem: 33SE from chapter: 2 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. The answer to “This exercise will lead you through a proof of Chebyshev's inequality. Let X be a continuous random variable with probability density function f(x). Suppose that P(X<0) = 0, so f(x) = 0 for x ? 0.a. Show that ________________b. Let k> 0 be a constant. Show that________________c. Use part (b) to show that P(X ? k) ? ?X / k. This is called Markov's inequality. It is true for discrete as well as for continuous random variables.________________d. Let Y be any random variable with mean ?Y and variance Let X = (Y ? ?Y)2. Show that ________________e. Let k > 0 be a constant. Show that ________________f. Use part (e) along with Markov's inequality to prove Chebyshev's inequality: P(|Y ? ?Y| ? k?Y) ? 1/k2.” is broken down into a number of easy to follow steps, and 125 words. Since the solution to 33SE from 2 chapter was answered, more than 280 students have viewed the full step-by-step answer.