(10 points) Suppose that a car can run for a random number of miles X before it's battery fails. X is a continuous variable with the following density: 15,000 x>0 f(x)= 15,000€ 0, I < 0 (a) Show that the expected life of the battery E(X) is 15,000 miles. (Consider using integration by parts.) (b) Determine P(X > 5000). (c) After having driven 5000 miles, suppose the battery has not failed. What is the chance that the battery will last the rest of your 10000 mile trip? I.e. determine P(X> 10000 | X > 5000). Show that this is equal to the chance P(X > 5000) : the unconditional probability that you can make it more than 5000 miles without battery failure.