Results 1 to 3 of 3

Math Help - Conditional expectation problem

  1. #1
    LHS
    LHS is offline
    Member
    Joined
    Feb 2009
    From
    Oxford
    Posts
    84

    Conditional expectation problem

    Hi,
    I'd be rather grateful if anyone could help me with this? Thanks

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    I don't know how to do this by using conditional expectations. My method would be to use the theory of Markov processes.

    The transition matrix for this process is \begin{bmatrix}0&\scriptstyle 1/3&0&1&0&0\\ \scriptstyle 1/2&0&1&0&\scriptstyle 1/2&0\\ 0&\scriptstyle 1/3&0&0&0&0\\ \scriptstyle 1/2&0&0&0&0&0\\ 0&\scriptstyle 1/3&0&0&0&0\\ 0&0&0&0&\scriptstyle 1/2&1\end{bmatrix}, where the (i,j)-element gives the transition probability from room j to room i. If  R is the 5x5 submatrix obtained by deleting the final row and column, then I-R = \begin{bmatrix}1&\scriptstyle -1/3&0&-1&0\\ \scriptstyle -1/2&1&-1&0&\scriptstyle -1/2\\ 0&\scriptstyle -1/3&1&0&0\\ \scriptstyle -1/2&0&0&1&0\\ 0&\scriptstyle -1/3&0&0&1\end{bmatrix}. The inverse of I-R is the so-called fundamental matrix of the process, (I-R)^{-1} = \begin{bmatrix}6&4&4&6&2\\ 6&6&6&6&3\\ 2&2&3&2&1\\ 3&2&2&4&1\\ 2&2&2&2&2\end{bmatrix}.

    According to the theory of absorbing Markov systems (see here, for example), the (i,j)-entry in the fundamental matrix gives the expected duration of being in state i, having started in state j. So if the mouse starts in room 1, then (reading off the numbers in the first column of the fundamental matrix) it can expect to spend a total of 6 minutes in each of rooms 1 and 2, 2 minutes in each of rooms 3 and 5, and 3 minutes in room 4, before disappearing into the fatal room 6. That gives it a total expected life of 6+6+2+3+2 = 19 minutes. (That is much longer than I would have guessed I hope I haven't got the arithmetic wrong.)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    LHS
    LHS is offline
    Member
    Joined
    Feb 2009
    From
    Oxford
    Posts
    84
    Wow! thank you for your detailed answer, I do believe 19 is correct. Surprising isn't it!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Conditional Expectation problem
    Posted in the Statistics Forum
    Replies: 1
    Last Post: August 14th 2011, 10:52 PM
  2. Replies: 0
    Last Post: July 22nd 2011, 01:39 AM
  3. Conditional Expectation Problem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 25th 2010, 09:50 AM
  4. Conditional Expectation Problem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: September 17th 2009, 03:28 AM
  5. Expectation & Conditional Expectation
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: February 1st 2009, 10:42 AM

Search Tags


/mathhelpforum @mathhelpforum