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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 | ISBN: 9780073401331 | Authors: William Navidi

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 \(x \leq 0\).

a. Show that \(\mu_{X}=\int_{0}^{\infty} x f(x) d x\).

b. Let \(k>0\) be a constant. Show that \(\mu_{X} \geq \int_{k}^{\infty} k f(x) d x=k P(X \geq k)\).

c. Use part (b) to show that \(P(X \geq k) \leq \mu_{X} / k\). 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 \(\mu_{Y}\) and variance \(\sigma_{Y}^{2}\). Let \(X=\left(Y-\mu_{Y}\right)^{2}\). Show that \(\mu_{X}=\sigma_{Y}^{2}\).

e. Let be \(k>0\)  a constant. Show that \(P\left(\left|Y-\mu_{Y}\right| \geq k \sigma_{Y}\right)=P\left(X \geq k^{2} \sigma_{Y}^{2}\right)\)

f. Use part (e) along with Markov's inequality to prove Chebyshev's inequality: \(P\left(\left|Y-\mu_{Y}\right| \geq k \sigma_{Y}\right) \leq 1 / k^{2}\)

Equation Transcription:

 

 

 

 

 

Text Transcription:

 X

f(x)

P(X<0)=0

f(x)=0

x leq 0

muX=integral_0^ infinity  xf(x)dx

mu_X geq integral_k^ infinity kf(x)dx=kP(X qec k)

P(X gec k)lec mu_X/k

Y

mu_Y

sigma_Y^2

X=(Y-mu_Y)^2

mu_X=sigma_Y^2

P(|Y-mu_Y| gec k sigma_Y)=P(X gec k^2 sigma_Y^2)

P(|Y-mu_Y|gec k sigma_Y) lec 1/k^2

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 \(x \leq 0\).a. Show that \(\mu_{X}=\int_{0}^{\infty} x f(x) d x\).b. Let \(k>0\) be a constant. Show that \(\mu_{X} \geq \int_{k}^{\infty} k f(x) d x=k P(X \geq k)\).c. Use part (b) to show that \(P(X \geq k) \leq \mu_{X} / k\). 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 \(\mu_{Y}\) and variance \(\sigma_{Y}^{2}\). Let \(X=\left(Y-\mu_{Y}\right)^{2}\). Show that \(\mu_{X}=\sigma_{Y}^{2}\).e. Let be \(k>0\) a constant. Show that \(P\left(\left|Y-\mu_{Y}\right| \geq k \sigma_{Y}\right)=P\left(X \geq k^{2} \sigma_{Y}^{2}\right)\)f. Use part (e) along with Markov's inequality to prove Chebyshev's inequality: \(P\left(\left|Y-\mu_{Y}\right| \geq k \sigma_{Y}\right) \leq 1 / k^{2}\)Equation Transcription: Text Transcription: Xf(x)P(X<0)=0f(x)=0x leq 0muX=integral_0^ infinity xf(x)dxmu_X geq integral_k^ infinity kf(x)dx=kP(X qec k) P(X gec k)lec mu_X/kYmu_Ysigma_Y^2X=(Y-mu_Y)^2mu_X=sigma_Y^2P(|Y-mu_Y| gec k sigma_Y)=P(X gec k^2 sigma_Y^2)P(|Y-mu_Y|gec k sigma_Y) lec 1/k^2” is broken down into a number of easy to follow steps, and 164 words. Since the solution to 33SE from 2 chapter was answered, more than 388 students have viewed the full step-by-step answer.

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