We suppose that the traffic at a point along a certain road can be described by a Poisson process with parameter λ = 2 per minute and that 60% of the vehicles are cars, 30% are trucks, and 10% are buses. We also suppose that the number K of persons in a single vehicle is a random variable whose function PK is
in the case of cars and trucks, respectively, and pk(k) = 1/50, for k = 1 , . . . ,50, in the case of buses.
(a) Calculate the variance of the number of persons who pass by this point in the course of a five-minute period.
(b) Given that five cars passed by the point in question over a five-minute period, what is the variance of the total number of vehicles that passed by that point during these five minutes?
Indication. We assume that the number of cars is independent of the number of trucks and buses.
(c) Calculate, assuming as in (b) the independence of the vehicles, the probability that two cars will pass by this point before two vehicles that are not cars pass by there.
(d) Suppose that actually
