The number of buses that arrive at a bus stop during a one-hourtime span can be modeled as a Poisson process with rate ? (seeRemark below). Now suppose a passenger has just arrived at the busstop and starts waiting. Let Y be the time (in unit of hours) sheneeds to wait to see the first bus.
(a) Is Y a discrete or continuous random variable? Find the setof all possible values of Y .
(b) For a possible value y of Y , find P(Y > y).
(c) Find the distribution (pdf/pmf) of Y and identify it as oneof the “named” distributions with corresponding parameter(s).
In the same setting as in the previous question, suppose we knowthere is exactly one bus arrival on a time interval [0, 1] (theunit is hour). Let Y be this single arrival time.
(d) Find the set of possible values of Y .
(e) For a possible value y of Y , find P(Y ? y).
(f) Find the pdf/pmf of Y and identify it as one of the “named”distributions with corresponding parameter(s).
