An automobile manufacturer has asked you to assist them with some financial analysis regarding the painting of automobiles. As part of the assembly process, each automobile passes through a paint booth where the final coat of paint is applied. The manufacturer paints two types of automobiles: single color and and two-tone. The assembly mix is 80% single color and 20% two-tone. Unfortunately, the painting process resulting in some cosmetic defects that must be repaired later in the assembly process. Assume the number of paint defects on an automobile follows a Poisson distribution with \lambda = 0.85 for single color and \lambda = 1.40 for two-tone. After assembly and final testing, if a vehicle has no paint defects, it is sent to the shipping area while vehicles with one or more paint defects go to the repair area. It costs the manufacturer $300 to repair a single color vehicle and $450 to repair a two tone.
a) What is the probability a single color vehicle will have exactly 2 paint defects?
b) What is the probability a two-tone vehicle will have exactly 1 paint defect?
c) What is the probability a single color vehicle will go to the repair area?
d) What proportion of all vehicles will go to the repair area?
e) Given that a vehicle is in the repair area, what is the probability that it is a two-tone?
f) You decide to model repair costs as a discrete random variable. Develop the probability mass function (pmf) for the repair cost per vehicle.
g) What is the expect repair cost, i.e., determine the expected value for your random variable in part f.