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Math Help - Help with a Markovian chain math problem?

  1. #1
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    Help with a Markovian chain math problem?

    Pls help solve this with a Markovian chain:
    Census studies from 1960 reveal that in France 70% of the daughters of working women also work, and 50% of the daughters of nonworking women work. Assume that this trend remains unchanged from one generation to the next,
    Set up a Markov chain by drawing the transition diagram and the transition matrix.
    That time 40% of French women worked. Determine the percentage of working women in the next two generations.
    In the long run what percentage of French women will work?
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  2. #2
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    Hello, inso!

    Census studies from 1960 reveal that in France 70% of the daughters of working women
    also work, and 50% of the daughters of nonworking women work.
    Assume that this trend remains unchanged from one generation to the next,

    (a) Set up a Markov chain by drawing the transition diagram and the transition matrix.
    w = work, .n = not work.


    . . \begin{array}{ccc}<br />
& _{0.3} & \\<br />
& \longrightarrow & \\<br />
0.7\:\circlearrowright\:\textcircled{w} & & \textcircled{n} \:\circlearrowleft\:0.5 \\<br />
& \longleftarrow & \\<br />
& ^{0.5} & \end{array}



    We have: . \begin{array}{c|cc|}<br />
& w & n \\ \hline<br />
w & 0.7 & 0.3 \\<br />
n & 0.5 & 0.5 \\ \hline \end{array}


    The transition matrix is: . A \;=\;\begin{pmatrix}0.7 & 0.3 \\ 0.5 & 0.5 \end{pmatrix}




    (b) At that time 40% of French women worked.
    Determine the percentage of working women in the next two generations.

    We have: . v \:=\:(0.4,0.6) at the first generation.

    Second generation: . v\cdot A \;=\;(0.4,0.6)\begin{pmatrix}0.7 & 0.3 \\ 0.5 & 0.5 \end{pmatrix} \;=\;(0.58, 0.42)
    . . Working women: 58%

    Third generation: . v\cdot A^2 \;=\;(0.4,0.6)\begin{pmatrix}0.7 & 0.3 \\ 0.5 & 0.5\end{pmatrix}^2 \;=\; (0.616, 0.384)
    . . Working women: 61.6%




    (c) In the long run what percentage of French women will work?
    We want the "steady state" vector.
    This is a vector u \:=\:(p,q) such that: . u\cdot A \:=\:u

    So we have: . (p,q)\begin{pmatrix}0.7&0.3\\0.5&0.5\end{pmatrix} \;=\;(p,q) \quad\Rightarrow\quad \begin{Bmatrix}0.7p + 0.5q \:=\:p \\ 0.3p + 0.5q \:=\:q \end{Bmatrix}


    Then we have: . \begin{array}{ccc}\text{-}0.3p + 0.5q &=& 0 \\ 0.3p - 0.5q &=& 0 \end{array}\quad\hdots .
    These two equatons are always equivalent.

    The second equation is always: . p + q \:=\:1


    Solve the system: . \begin{array}{ccc}0.3p - 0.5p &=&0 \\ p + q &=& 1 \end{array} \quad\Rightarrow\quad (p,q) \:=\:(0.625, 0.375)


    Eventually, 62.5% of the women will be working.

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  3. #3
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    10ks sooo verry much!!!
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