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Let X and Y be random variables, and a and b be

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 27E Chapter 2.6

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 27E

Let X and Y be random variables, and a and b be constants.

a. Prove that \(\operatorname{Cov}(a X, b Y)=a b \operatorname{Cov}(X, Y)\).

b. Prove that if \(a>0\) and \(b>0\), then \(\rho_{a X, b Y}=\rho_{X, Y}\). Conclude that the correlation coefficient is unaffected by changes in units.

Equation Transcription:

Text Transcription:

Cov(aX,bY)=ab Cov(X,Y)

a>0

b>0

rho_{aX,bY}=rho_{X,Y}

Step-by-Step Solution:

Solution :

Step 1 of 2:

Let X and Y be the random variable.

Our goal is :

a). We need to prove that Cov(aX,bY)=abCov(X,Y).

b). We need to prove that if a>0 and b>0,then  .

a).

Now we need to prove that Cov(aX,bY)=abCov(X,Y)

Here a and b are the constants.

Here

and

Then,

Here

Therefore


Step 2 of 2

Chapter 2.6, Problem 27E is Solved
Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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Let X and Y be random variables, and a and b be