Problem 13E
Refer to Exercise 12. Let Z = X + Y represent the total number of repairs needed.
a. Find μZ.
b. Find σZ.
c. Find P(Z = 2).
Exercise 12
Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a onehour time interval. The joint probability mass function of X and Y is as follows:
a. Find the marginal probability mass function px (x).
b. Find the marginal probability mass function py (y).
c. Find μx.
d. Find μy.
e. Find σx.
f. Find σy.
g. Find Cov(X, Y).
h. Find ρx,y.
Solution:
Step1 of 4:
Here automobile engines and transmissions are produced on assembly lines. They are inspected for defects. Those with defects are replaced. Let X is the number of engines and Y is the number of transmissions. The joint probability mass function of X and Y is given as,
Y  
X 
0 
1 
2 
3 

0 
0.3 
0.10 
0.07 
0.03 
0.33 
1 
0.12 
0.16 
0.08 
0.04 
0.40 
2 
0.02 
0.06 
0.08 
0.04 
0.20 
3 
0.01 
0.02 
0.02 
0.02

0.07 
0.28 
0.34 
0.25 
0.13 
We can find the marginal probability mass function of X and Y by summing along the row and column of joint pmf respectively.
Let Z= X+Y represents the total number of repairs needed. We have to find
 = E(Z)
 = Standard deviation of Z
(c) P(Z=2)