Problem 9. Each vehicle arriving at a toll booth is either a car or a truck. Cars arrive as a Poisson process with rate A = 3 per minute. Independently of the cars, trucks arrive as a Poisson process with rate p. = 1 per minute. {a} Consider the rst vehicle to arrive after time t = 0. What is the probability that this vehicle arrives after time t = 1 and is a truck? (b) Each car has to pay a toll of 3 dollars and each truck has to pay a toll of 5 dollars. Let M be the amount of dollars collected from these tolls in time interval [15, 25]. Find the variance of M. [c) What is, approximately, the probability that in the rst 100 minutes the number of cars that have arrived is more than the number of trucks by at least 230