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Math Help - Discrete Time Markov Chain

  1. #1
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    Discrete Time Markov Chain

    Hey Guys,

    A quick question I was wondering if you could help me or help me try and understand the way about approaching this question...

    Let {Xt}t≥0 be a two-state Markov chain with state space S = {0, 1}, transition matrix:

    P = \left(\begin{array}{cc}1-p&p\\q&1-q\end{array}\right)

    and initial distribution  \pi_0 = ({\pi_0(0),\pi_0(1)})

    Define the New Stochastic Processes {Yt}t≥1 and {Zt}t≥1 as:

    Y_t = X_t + X_{t-1} and,

    Z_t = 10X_t + X_{t-1}

    a) What are the State Spaces for these New Stochastic Processes?

    I have tried using the transition matrix and inputting the values of S = {0,1} into {Yt} and {Zt} but I'm not sure if I am on the right path.. Any Suggestions??

    Thanks
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  2. #2
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    Please somebody help!!!

    So I don't feel so alone with Markov Chains?!
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  3. #3
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    The state space is the set of possible values which the random variable can take on. In this example if p,q are both non zero then there is some chance of going from any state to any other state in { 0,1}.

    Therefore:
    <br />
Y_t = X_t + X_{t-1}<br />

    can be 0 when X_t = X_{t-1} = 0
    or 1 when X_t = 0 \text{ and } X_{t-1} = 1 \text{ or } X_t = 1 \text{ and } X_{t-1} = 0
    or 2 when X_t = 1 \text{ and } X_{t-1} = 1

    Use the same reasoning to determine what states Z_t can be in. Can it achieve the value 11?
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  4. #4
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    Thanks for helping to clear up my interpretation of the State Space!

    I'm assuming that with  Z_t = 10X_t + X_{t-1}

     0 when  10X_t = X_{t-1} = 0
     1 when  10X_t = 0 and  X_{t-1} = 1
     10 when  10X_t = 1 and  X_{t-1} = 0
     11 when  10X_t = 1 and  X_{t-1} = 1

    So would I be right in saying that  S_z = \{0,1,10,11\}
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