Here is the problem:
And here is what I tried to do:
I tried doing it in my calculator and got that the answer is 40 and 160 as you keep multiplying PX. Can anyone point out what I'm doing wrong or help me?
Thanks.
Edit for readability:
Problem Statement:
A college dormitory houses 200 students. Those who watch an hour or more of TV on any day always watch for less than an hour the next day. One-fourth of those who watch TV for less than an hour one day will watch an hour or more the next day. Half of the students watch TV for an hour or more today.
a) How many will watch TV for an hour or more tomorrow?
b) In 2 days?
c) Find the steady state matrix for populations described in this problem.
Work:
$
P = \stackrel{>1 \hphantom{XX} <1}{\begin{bmatrix}
0 & 0.25\\
1 & 0.75
\end{bmatrix}} {>1 \atop <1}
\quad
x = \begin{bmatrix}
100\\
100
\end{bmatrix}
{>1 \atop <1}
$
$
Px = \begin{bmatrix}
0 & 0.25\\
1 & 0.75
\end{bmatrix}
\begin{bmatrix}
100\\
100
\end{bmatrix}
=
\begin{bmatrix}
25\\
175
\end{bmatrix}
{>1 \atop }
$
a) 25.
$
P^{\,2}x = \begin{bmatrix}
0 & 0.25\\
1 & 0.75
\end{bmatrix}
\begin{bmatrix}
25\\
175
\end{bmatrix}
=
\begin{bmatrix}
43.75\\
156.25
\end{bmatrix}
{>1 \atop <1}
$
b) $ 43.75 \sim 44 $.
Steady state $P\,\overline{x} = \overline{x}$
\begin{bmatrix}
0 & 0.25\\
1 & 0.75
\end{bmatrix}
\begin{bmatrix}
x_{1}\\
x_{2}
\end{bmatrix}
\begin{bmatrix}
x_{1}\\
x_{2}
\end{bmatrix}
$
\Rightarrow
$
\begin{bmatrix}
0.25 x_{2}\\
x_{1}+0.75 x_{2}
\end{bmatrix}
$
\Rightarrow
$
\begin{bmatrix}
25 \\
175
\end{bmatrix}
$
\boxed{\begin{align*}
0.25 x_{2} &= x_{1}\\
x_{1} + 0.75 x_{2} &= x_{2}
\end{align*}}
$
Best Answer
The steady state is the population split that is an eigenvector of the transition matrix with eigenvalue 1, that is:
$$\left(\begin{matrix} 0 & 1/4 \\ 1 & 3/4 \end{matrix}\right) \left( \begin{matrix} x \\ y \end{matrix} \right) = \left( \begin{matrix} x \\ y \end{matrix} \right)$$
This leads you to the equation
$$y = 4x$$
and you also know that
$$x + y = 200$$
and solving these jointly gives $x=40$, $y=160$.