A bus with passenger capacity K stops at stations numbered 0,1,2,3 ... on an unlimited route. Denote by the number of passengers waiting to board the bus at station n. Assume that {, 0} are iid random variables with common distribution = ( = ), = 0,1,2, ... Each passenger on the bus leaves at any given station with constant probability p, independently from the other passengers. Once people have exited, as many passengers waiting at a given station will board as there are free places on the bus available. Denote by the number of passengers on the bus after leaving station n. Show that {, 0} is a DTMC and calculate its transition probability matrix.