Suppose the numbers of families that check into a hotel on succes- sive days are independent Poisson random variables with mean λ. Also suppose that the number of days that a family stays in the hotel is a geometric random variable with parameter p,0 p 1. (Thus, a family who spent the previous night in the hotel will, independently of how long they have already spent in the hotel, check out the next day with probability p.) Also suppose that all families act independently of each other. Under these conditions it is easy to see that if Xn denotes the number of families that are checked in the hotel at the beginning of day n then {Xn, n 0} is a Markov chain. Find

(a) the transition probabilities of this Markov chain;

(b) E[Xn|X0 = i];

(c) the stationary probabilities of this Markov chain.