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# In a piston assembly, the specifications for the clearance

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

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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

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In a piston assembly, the specifications for the clearance