Question

Use the matrix of transition probabilities P and initial state matrix X_0 to find the state matrices X_1, X_2, and X_3. P = [0.6 0.2 0.1 0.3 0.7 0.1 0.1 0.1 0.8], X_0 = [0.1 0.2 0.7] X_1 = [] X_2 = [] X_1 = []

Answer 1

When we multiply the initial state matrix X₀ with the transition matrix P, we get state matrix X₁.
$\dpi{100} X_{1}=PX_{0}$

Again multiplying with the transition matrix, we get state matrix X₂ and so on.
$\dpi{100} X_{2}=PX_{1}$
$\dpi{100} X_{3}=PX_{2}$

$\dpi{100} \therefore X_{1}=\begin{bmatrix} 0.6 & 0.2 & 0.1\\ 0.3& 0.7 &0.1 \\ 0.1& 0.1 & 0.8 \end{bmatrix}\begin{bmatrix} 0.1\\ 0.2\\ 0.7 \end{bmatrix}= \begin{bmatrix} 0.17\\ 0.24\\ 0.59 \end{bmatrix}$

$\dpi{100} X_{2}=\begin{bmatrix} 0.6 & 0.2 & 0.1\\ 0.3& 0.7 &0.1 \\ 0.1& 0.1 & 0.8 \end{bmatrix}\begin{bmatrix} 0.17\\ 0.24\\ 0.59 \end{bmatrix}= \begin{bmatrix} 0.209\\ 0.278\\ 0.513 \end{bmatrix}$