Results 1 to 6 of 6

Math Help - Markov chain

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    24

    Question Markov chain

    Hi, can you help me with this problem, please?
    I have tried but I am not sure what to do exactly...
    Yn - sum of n independent rolls of a fair die.
    Find lim P{Yn is a multiple of 13), n->inf

    How can I define the appropriat Markov chain?
    I know that after defining it I can use the limiting probability

    If you can help me with this will be greatly appreciated.

    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by bamby View Post
    Hi, can you help me with this problem, please?
    I have tried but I am not sure what to do exactly...
    Yn - sum of n independent rolls of a fair die.
    Find lim P{Yn is a multiple of 13), n->inf

    How can I define the appropriat Markov chain?
    I know that after defining it I can use the limiting probability

    If you can help me with this will be greatly appreciated.

    Thank you!
    Let the state be X_n be the value of Y_n \text{ mod } 13

    and distribution vector (x_{n,0},x_{n,1},\ ..\ ,x_{n,12})

    where x_{n,i}=Pr(Y_n \equiv i \text{ mod } 13).

    Now write the transition matrix for this process.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    24
    But that is the point - I don't know the transititon matrix.
    I know that the multiples of 13 are i mod 13
    Please, help me with this
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by bamby View Post
    But that is the point - I don't know the transititon matrix.
    I know that the multiples of 13 are i mod 13
    Please, help me with this
    The point is that if you throw a single dice you may assume that each of the faces comes up with probability 1/6 and the faces show the values 1,2,3,4,5,6.

    From that you can work out the transition matrix.

    So if X_n=8, then:

    X_{n+1}=9 with prob 1/6,

    X_{n+1}=10 with prob 1/6,

    X_{n+1}=11 with prob 1/6,

    X_{n+1}=11 with prob 1/6,

    X_{n+1}=0 with prob 1/6,

    X_{n+1}=1 with prob 1/6.

    CB
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Sep 2008
    Posts
    24
    So, I have got for the transition matrix

    A=[0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0;
    0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0;
    0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0;
    0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0;
    0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0;
    0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0;
    0 0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6;
    1/6 0 0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6;
    1/6 1/6 0 0 0 0 0 0 0 1/6 1/6 1/6 1/6;
    1/6 1/6 1/6 0 0 0 0 0 0 0 1/6 1/6 1/6;
    1/6 1/6 1/6 1/6 0 0 0 0 0 0 0 1/6 1/6;
    1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0 0 1/6;
    1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0 0]

    and for the limit I raised it A^30 and got 0.0769
    Is that correct?

    Thank you so much!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by bamby View Post
    So, I have got for the transition matrix

    A=[0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0;
    0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0;
    0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0;
    0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0;
    0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0 0;
    0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6 0;
    0 0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6 1/6;
    1/6 0 0 0 0 0 0 0 1/6 1/6 1/6 1/6 1/6;
    1/6 1/6 0 0 0 0 0 0 0 1/6 1/6 1/6 1/6;
    1/6 1/6 1/6 0 0 0 0 0 0 0 1/6 1/6 1/6;
    1/6 1/6 1/6 1/6 0 0 0 0 0 0 0 1/6 1/6;
    1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0 0 1/6;
    1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0 0]

    and for the limit I raised it A^30 and got 0.0769
    Is that correct?

    Thank you so much!
    Well it looks OK (but it can be dificult to be sure as there are two conventions for the matrix).

    An alternative is to look at the solution of the equation:

    xA=x

    then the answer required is x_1.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Markov Chain of random variables from a primitive markov chain
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 19th 2011, 09:12 AM
  2. Markov Chain Help
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: June 28th 2010, 08:37 AM
  3. Markov Chain
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: December 12th 2009, 05:52 PM
  4. Markov Chain HELP!!!!
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 9th 2009, 10:28 PM
  5. Replies: 2
    Last Post: October 28th 2008, 07:32 PM

Search Tags


/mathhelpforum @mathhelpforum