(Solved): 2. (50 points) Just like all other cars, a car has four tires and a spare one in its trunk. The lif ...

2. (50 points) Just like all other cars, a car has four tires and a spare one in its trunk. The lifetimes of the four tires at the beginning of a long trip are identically (meaning has the same distribution) and independently distributed exponential random variables with a mean of 5.000 kilometers. The spare tire, on the other hand, has an exponential lifetime with a mean of 400 kilometers. a) (10 points) What is the rate that you change the left front tire in 10,000 kilometers? What is the rate that you change one of the tires in 15,000 kilometers? b) (10 points) What is the probability that the car travels without changing a tire for more than 8,000 kilometers? Hint: Let T: kilometers traveled without changing a tire and T., T. TT, and T. are lifetimes of four tires and the spare tire, respectively. Spring 2020 c) (10 points) Given that the car already traveled 7,000 kilometers, what is the probability that the car travels without changing a tire for more than 12,000 kilometers? Note: Consider the hint in part b). d) (10 points) Given that one of the tires failed after traveling 3,000 kilometers, find the probability that the car travels another 7,000 kilometers without going to a repair shop. Let Ts: kilometers traveled until the first failure, Tx: kilometers traveled until the second failure Hint: What is the distribution of T:? What is the distribution of T:? el (10 points) Find the average trip length the car travels with the original four tires. Find the average trip length the car travels with the spare tire. Explain and compare your results.