Question

Let {Xn : n ? 0} be a Markov chain with state space {0, 1, 2, 3} and transition matrix P = ? 1/4 0 1/2 1/4 ? ? 0 1/5 0 4/5 ? ? 0 1 0 0 ? ? 1/3 1/3 0 1/3 ? (a) Draw a state space diagram for the process {Xn : n ? 0}. (b) Compute P(X17 = 1, X15 = 0, X14 = 3, X12 = 2 | X10 = 3), justifying each step carefully. A new process {Zn : n ? 0} is defined by Zn = 0 if Xn = 0 or 1, and Zn = Xn if Xn = 2 or 3. (c) Write down the state space of the process {Zn : n ? 0}. (d) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 2). (e) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 3). (f) Is Z0, Z1, Z2, ... a Markov chain? Why? HINT: to find these probabilities, think what the equivalent value for Xn would be. For example, if Zn = 2 then we must have that Xn = 2. By doing this, you can use the probabilities in the transition matrix for {Xn : n ? 0}.

          Let {Xn : n ? 0} be a Markov chain with state space {0, 1, 2, 3} and transition matrix
P = 
? 1/4  0   1/2  1/4 ?
?  0   1/5  0   4/5 ?
?  0    1   0    0  ?
? 1/3  1/3  0   1/3 ?

(a) Draw a state space diagram for the process {Xn : n ? 0}.
(b) Compute P(X17 = 1, X15 = 0, X14 = 3, X12 = 2 | X10 = 3), justifying each step carefully.

A new process {Zn : n ? 0} is defined by Zn = 0 if Xn = 0 or 1, and Zn = Xn if Xn = 2 or 3.

(c) Write down the state space of the process {Zn : n ? 0}.
(d) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 2).
(e) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 3).
(f) Is Z0, Z1, Z2, ... a Markov chain? Why?
HINT: to find these probabilities, think what the equivalent value for Xn would be. For example, if Zn = 2 then we must have that Xn = 2. By doing this, you can use the probabilities in the transition matrix for {Xn : n ? 0}.

Let Xn : n ? 0 be a Markov chain with state space 0, 1, 2, 3 and transition matrix
P =
? 1/4 0 1/2 1/4 ?
? 0 1/5 0 4/5 ?
? 0 1 0 0 ?
? 1/3 1/3 0 1/3 ?

(a) Draw a state space diagram for the process Xn : n ? 0.
(b) Compute P(X17 = 1, X15 = 0, X14 = 3, X12 = 2 | X10 = 3), justifying each step carefully.

A new process Zn : n ? 0 is defined by Zn = 0 if Xn = 0 or 1, and Zn = Xn if Xn = 2 or 3.

(c) Write down the state space of the process Zn : n ? 0.
(d) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 2).
(e) Find P(Zn+1 = 2 | Zn = 0, Zn-1 = 3).
(f) Is Z0, Z1, Z2, ... a Markov chain? Why?
HINT: to find these probabilities, think what the equivalent value for Xn would be. For example, if Zn = 2 then we must have that Xn = 2. By doing this, you can use the probabilities in the transition matrix for Xn : n ? 0.

Added by Heather B.

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