Question

A state vector X for a four-state Markov chain is such that the system is four times as likely to be in state 3 as in 1, is not in state 4, and is in state 2 with probability 0.2. Find the state vector X .

Answer 1

The components of a state vector must add up to 1. If $x_i$ denotes the probability that the system is in state $i$, and given the conditions above, then you have

$\begin{cases}x_3=4x_1\\x_4=0\\x_2=0.2\\x_1+x_2+x_3+x_4=1\end{cases}$

The last equation reduces to

$x_1+0.2+4x_1+0=1\implies 5x_1=0.8\implies x_1=0.16\implies x_3=4(0.16)=0.64$

So the state vector is

$\mathbf x=\begin{bmatrix}0.16\\0.2\\0.64\\0\end{bmatrix}$