3. Suppose you have two fair coins. Each coin has one side labelled "1" and the other sided labelled "2". You toss both coins at the same time, repeatedly. Let Qn be the sum of the two numbers the (n + 1) time both coins are tossed. Consider the stochastic process X = (Xo, X1, X2, ...) as defined in each of the separate cases A-D below. Answer the following for each case: (i) Does X form a Markov chain? If so, find the corresponding state space S, the initial distribution w, and the one-step transition matrix P matrix. (ii) If not, find a new sequence that forms a Markov chain and has enough infor- mation to recover the original sequence. Case A. Define Xn := Qn for each n 2 0. Case B. Define X, := max Q; for each n 2 0, that is, Xo, X1, ... is the sequence of 1=0, ...,n the running-maximum of the Q's. Case C. Let (Xn)nez, be the sequence of running totals, that is, define Xn := > Qi i=0 Case D. The sequence (Xn)nez, is defined by Xn = Qn + Qn-1 for n = 1, 2