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Let the joint pmf of X and Y be (a) Are X and Y

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman ISBN: 9780321923271 41

Solution for problem 7E Chapter 4.2

Probability and Statistical Inference | 9th Edition

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Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

Probability and Statistical Inference | 9th Edition

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Problem 7E

PROBLEM 7E

Let the joint pmf of X and Y be

(a) Are X and Y independent?

(b) Calculate Cov(X,Y) and ρ.

This exercise also illustrates the fact that dependent random variables can have a correlation coefficient of zero.

Step-by-Step Solution:

Answer

Step 1 of  4

The given pmf is (x,y)S={(0,0), (1,1), (1,-1), (2,0)}

Arrange the given values into tabular form

0

1

2

-1

0

1/4

0

1/4

0

1/4

0

1/4

2/4

1

0

1/4

0

1/4

1/4

2/4

1/4

1

Step 2 of  4

a) If X and Y are independent then it satisfies E(XY)=E(X)E(Y)

     E(X)=

            = 0(1/4)+1(2/4)+2(1/4)

            = 4/4

            =1

    E(Y)=

            = -1(1/4)+0(2/4)+1(1/4)

            = 1/4-1/4

            =0

 

Step 3 of 4

Chapter 4.2, Problem 7E is Solved
Step 4 of 4

Textbook: Probability and Statistical Inference
Edition: 9
Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman
ISBN: 9780321923271

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Let the joint pmf of X and Y be (a) Are X and Y

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