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 ).
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 |
|
