I am wondering how can I find an initial state vector for this problem.
If the air quality is good one day, it has 95% chance it will be good the next day. If the air quality is bad one day it has 45% chance it will be bad the next day.
From this information I constructed the transition Matrix P:
$$P = \begin{bmatrix}0.95 & 0.55 \\ 0.05 & 0.45\end{bmatrix}$$
then the question asks, if the weather is good today, what is the chance it will be good two days from today:
I assume to find this I would need to calculate:
$x(t) = P^t * x_o$ , where $x_o$ is the initial state vector, therefore
$x(2) = P^2 * x_o$ But how do I find this initial state vector?
Best Answer
The initial state vector is that today the weather is good, so it is $x_0=(1, 0)$. You can also do it as follows. Let $x_0$ be the weather today, $x_1$ tomorrow and $x_2$ two days from now. The problem specifies that $x_0=x_2=$good weather(g). Denote bad weather with (b). So the possible transitions are $gbg$ or $ggg$ with probabilities: $$P(x_0=g, x_1=b, x_2=g)=0.05\cdot0.55=0.0275$$ and $$P(x_0=g, x_1=b, x_2=g)=0.95\cdot0.95=0.9025$$ So the probability you are looking for is $$P(x_0=g, x_1=b, x_2=g)+P(x_0=g, x_1=b, x_2=g)=0.0275+0.9025=0.93$$