Vehicles pass a certain marker on a highway in accordance with a Poisson process at the rate of 48 per hour; 25 % of the vehicles on the road are trucks and 75 % are cars. Suppose you observe this marker at a particular point of time.
(a) What is the probability that exactly five vehicles will pass in the first 15 minutes?
(b) Given that exactly two trucks passed in the first 5 minutes, what is the probability that the fourth truck will pass in the first 10 minutes?
(c) Given that exactly two trucks passed in the first 5 minutes, what is the expected value and the variance of the number of vehicles that passed in the first 10 minutes?