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In a piston assembly, the specifications for the clearance

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

Solution for problem 4E 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 4E

In a piston assembly, the specifications for the clearance between piston rings and the cylinder wall are very tight. In a lot of assemblies, let X be the number with too little clearance and let Y be the number with too much clearance. The joint probability mass function of X and Y is given in the table below:

a. Find the marginal probability mass function of X.

b. Find the marginal probability mass function of Y .

c. Are X and Y independent? Explain.

d. Find μX and μY .

e. Find σX and σY .

f. Find Cov(X, Y ).

g. Find ρ(X, Y ).

Step-by-Step Solution:

Solution :

Step 1 of 7:

Given, the number with too little clearance is X and

The number with too much clearance is Y.

Then the joint probability mass function of X and Y is given in the table below.

                      y

x

0

1

2

3

0

0.15

0.12

0.11

0.1

1

0.09

0.07

0.05

0.04

2

0.06

0.05

0.04

0.02

3

0.04

0.03

0.02

0.01

Our goal is:

a). Find the probability mass function of X.        

b). Find the probability mass function of Y.

c). Are X and Y are not independent? Explain.

d). Find mean and .

e). Find .

f). Find Cov (X,Y).

g). Find .

a).

Now we have to find the probability mass function of X.

Here we are summing all the rows to find a probability mass function.

Then the  table is given below.

                      y

Total

x

0

1

2

3

0

0.15

0.12

0.11

0.1

0.15+0.12+0.11+0.1

1

0.09

0.07

0.05

0.04

0.09+0.07+0.05+0.04

2

0.06

0.05

0.04

0.02

0.06+0.05+0.04+0.02

3

0.04

0.03

0.02

0.01

0.04+0.03+0.02+0.01

Then,

 

                      y

Total

x

0

1

2

3

0

0.15

0.12

0.11

0.1

0.48

1

0.09

0.07

0.05

0.04

Step 2 of 1

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

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In a piston assembly, the specifications for the clearance