The service department of a small car dealership employs 6 technicians that serv

Question

The service department of a small car dealership employs 6 technicians that service customers’ cars. With vacation days, sick days and other time off, not all of the technicians come to work each day. The dealership sells work by the hour, and the average number of hours per technician that are produced each day also varies. For any particular day, let X= the number of technicians working on that day and Y= the average number of hours produced per technician on that day. Based on the dealership’s payroll system, data have been used to determine that the joint probability mass function for X and Y is as given below.

1. Find the value of A in the pmf above.
2. Find the marginal distribution of X and give correct notation for the distribution.
3. Find the mean and variance of X.
4. Find the marginal distribution of Y and give correct notation for the distribution.
5. Find the mean and variance of Y.
6. Are X and Y independent? Explain how you can tell. (No credit without a valid explanation.)

x=4 x=5 x=6 Y=7 .1 .15 y=8 .35 .1 y=9 .05 A Y=10 .1

Explanation / Answer

1. A=1-.1-.15-.35-.1-.05-.1=0.15

2.

3. mean=5.1

variance=0.39

4

5. mean= 8.15

variance= 0.828

6. x and y are independent if P(X=x,Y=y)=P(X=x)*P(Y=y)

P(X=5,Y=7)=0.1

P(X=5)*P(Y=7)=0.6*0.25=0.15

therefore x and y are not independent.

x=4 x=5 x=6 0.15 0.6 0.25

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