Question 2 (30 pts). Suppose that the number of vehicles passing by the NUS entrance A is modelled by a homogeneous Poisson process with the rate of 10 vehicles per minute. Each arriving vehicle is either a lorry or a car. It is a lorry with probability 0.15 and it is a car with probability 0.85. (a) (b) (C) (d) (e) (f) In addition, the types of the arriving vehicles are independent. (3 pts) Find the probability that in 3 mins, at least one lorry passes by the NUS entrance A. (3 pts) Suppose that during the first minute of observation, exactly 12 lorries have passed by. Find the expected number of the cars passing the entrance during the same period. (4 pts) Suppose that during the rst minute of observation, exactly 15 vehicles have passed by. Find the probability that among these vehicles, there were exactly 3 lorries and 12 cars. (4 pts) What is the probability that the rst four arriving vehicles are all cars? (10 pts) Each car contains C many people, where C is a discrete RV with probability distri- bution Pr(C = 1) = 0.5,Pr(C = 2) = 0.3,Pr(C = 3) = 0.2. Each lorry contains L many people, where Pr(L = 2) = 0.3, Pr(L = 10) = 0.7. The numbers of people in all the vehicle are independent of the to others. Let Y be the total number of people passing by the NUS entrance A in a vehicle, during a 12 minutes period. (a) (4 pts) Compute 1E[Y]. (b) (6 pts) Compute Var(Y). (6 pts) Each car's driver is a male with probability 0.6, and is female with the complimentary probability 0.4. Each lorry's driver is a male with probability 0.8, and is a female with the complimentary probability 0.2. The driversJ genders are jointly independent. Find the probability that the rst arriving male car driver passes by the NUS entrance A before the rst arriving female lorry driver