Question

NI 100). Puu the profit of this him. 6. Use equation (8.2) and the data for P and xº given in example 8.10 to find the regional population distribution after three time periods.

Answer 1

We have the Markov's equation

$x^{n}=P^{n}x^{0}$

Here $x^{0}$=$\begin{bmatrix} 5\\ 10\\ 6\end{bmatrix}$

$P=\begin{bmatrix} .80 & .15& .05\\ .1& .7& .05\\ .1 & .15& .9\end{bmatrix}$

We need to find,$x^{3}=P^{3}x^{0}$

$x^{3}=\begin{bmatrix} .80 & .15& .05\\ .1& .7& .05\\ .1 & .15& .9\end{bmatrix}^{3}\begin{bmatrix} 5\\ 10\\ 6\end{bmatrix}$

=$\begin{bmatrix} .5605& .2756&.1278 \\ .1837&.3947 & .1098\\ .2557& .3296 & .7622 \end{bmatrix}$$\begin{bmatrix} 5\\ 10\\ 6\end{bmatrix}$

=$\begin{bmatrix} 6.31\\ 5.52\\ 9.14\end{bmatrix}$​​​​​​

Region 1=6.31,Region 2=5.52,Region 3=9.14

We see that region 2 is losing people because the value has decreased while region 1 and 3 are gaining people because their corresponding values are increasing.

Thank you for the question.