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Math Help - markov chain-fair die

  1. #1
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    markov chain-fair die

    A fair die is thrown repeatedly. Let S_{n} be the sum of the outcomes, and let R_{n} be the remindeer when S_{n} is divided by 4(that is R_{n} is the sum of the first n throws reduced modulo 4).
    a) Show that R_{n} ia a Markov chain on state space {0,1,2,3}.
    b) what are the transitoin probabilities for this chain?
    Can you please give me numerical example of R_{n} to see what is going on in this chain as I find it difficult to understand the question.
    thanks for any help.
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    Quote Originally Posted by charikaar View Post
    A fair die is thrown repeatedly. Let S_{n} be the sum of the outcomes, and let R_{n} be the remindeer when S_{n} is divided by 4(that is R_{n} is the sum of the first n throws reduced modulo 4).
    a) Show that R_{n} ia a Markov chain on state space {0,1,2,3}.
    b) what are the transitoin probabilities for this chain?
    Can you please give me numerical example of R_{n} to see what is going on in this chain as I find it difficult to understand the question.
    thanks for any help.
    I don't know if this is what you mean by a numerical example but here it goes. Suppose you rolled 4,2,6,1 then R looks like (for n=1,2,3,4) 0,2,0,1.

    a) should be relatively simple (use the Markov property of the S_n) and for b), think about it in this way, suppose you are at 0 and want to go to 1. You have to roll either a 1, or a 5 (both give remainder 1). So from 0 to 1 has probability 1/3. Now try to figure the other ones out using the same method.

    Hope this helps.
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