Markov chain problems, absorbing states and eigenvalues Problem set A Answers

\(T=\begin{pmatrix}0.6 & 0.2 & 0.1 \\ 0.4 & 0.6 & 0 \\ 0 & 0.2 & 0.9\end{pmatrix}\)

Three vertices; loop at A 0.46; A to B 0.54; B to A 0.28; loop at B 0.22; B to C 0.5; C to B 0.64; loop at C 0.36.

\(T=\begin{pmatrix}0.85 & 0.35 \\ 0.15 & 0.65\end{pmatrix}\)

1. (a) 0.264
2. (b) 0.449

1. (a) 0.85x + 0.3y = x; 0.15x + 0.7y = y; x+y=1
2. (b) x = 2/3, y = 1/3

1. (a) \(T=\begin{pmatrix}0.7 & 0.1 & 0 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.3 & 0.8\end{pmatrix}\)
2. (b) (156, 333, 511)
3. (c) (111, 333, 556)

0.38

0.38

1. (a) \(T=\begin{pmatrix}1 & 0.55 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.55 & 0 & 0 & 0 \\ 0 & 0.45 & 0 & 0.55 & 0 & 0 \\ 0 & 0 & 0.45 & 0 & 0.55 & 0 \\ 0 & 0 & 0 & 0.45 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.45 & 1\end{pmatrix}\)
2. (b) 0.286

1. (a) Eigenvalues 1 and 0.58; eigenvectors \(\begin{pmatrix}2 \\ 5\end{pmatrix}\) and \(\begin{pmatrix}1 \\ -1\end{pmatrix}\) (or multiples).
2. (b) \(P=\begin{pmatrix}2 & 1 \\ 5 & -1\end{pmatrix},\ D=\begin{pmatrix}1 & 0 \\ 0 & 0.58\end{pmatrix}\)

1. (a) \(f_n=0.7+0.2(0.4)^{n}\)
2. (b) 0.7

1. (a) 1, 0.5, 0.25
2. (b) A multiple of \(\begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}\)
3. (c) \(\begin{pmatrix} \frac{1}{5} \\ \frac{3}{10} \\ \frac{1}{2} \end{pmatrix}\)