# Math Help - Help with a Markovian chain math problem?

1. ## 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?

2. 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.

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

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

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.

3. 10ks sooo verry much!!!