A shuttle bus with infinite capacity stops at bus stops numbered 0, 1, 2, . . . on its infinite route. Let Yn be the number of people who board the bus at stop n. Assume that {Yn, n ? 0} is an i.i.d. sequence of random variables with common PMF: pk = P[Yn = k], k = 0, 1, 2 . . . Every passenger who is on the bus when the bus arrives at stop n has a probability p of alighting at that stop. The passengers behave independently of each other. Let Xn be the number of passengers on the bus when it leaves the nth stop. Show that {Xn, n ? 0} is a DTMC. Display its transition probability matrix.

A shuttle bus with infinite capacity stops at bus stops numbered 0,1,2, . .. on its infinite route. Let Y\" be the number of people who board the bus at stop n. Assume that {Ym n. 2 0} is an i.i.d. sequence of random variables with common PM F: pk = ]P'[Y = k], k = l], 1, 2 . .. Every passenger who is on the bus when the bus arrives at stop 11 has a probability p of alighting at that stop. The passengers behave independently of each other. Let X\" be the number of passengers on the bus when it leaves the nth stop. Show that {an 2 0} is a DTMC. Display its transition probability matrix