Transition matrices and diagrams Problem set B Answers

1. (a) \(\begin{pmatrix} 0.58 & 0.09 \\ 0.42 & 0.91 \end{pmatrix}\)
2. (b) \(\begin{pmatrix} 0.24 & 0.67 \\ 0.76 & 0.33 \end{pmatrix}\)

\(\begin{pmatrix} 0.38 & 0.11 & 0.29 \\ 0.47 & 0.64 & 0.21 \\ 0.15 & 0.25 & 0.5 \end{pmatrix}\)

\(\begin{pmatrix} 0.41 & 0.33 & 0.06 & 0.13 \\ 0.14 & 0.28 & 0.43 & 0.21 \\ 0.26 & 0.09 & 0.17 & 0.52 \\ 0.19 & 0.3 & 0.34 & 0.14 \end{pmatrix}\)

1. (a) Loops: P 0.81, Q 0.54. Arrows: P to Q 0.19, Q to P 0.46.
2. (b) Loops: A 0.58, B 0.23, C 0.88. Arrows: A to B 0.42, B to A 0.27, B to C 0.5, C to A 0.12.

\(\begin{pmatrix} 0.41 & 0.18 \\ 0.59 & 0.82 \end{pmatrix}\)

\(\begin{pmatrix} 1 & 0.3 & 0.1 \\ 0 & 0.5 & 0.45 \\ 0 & 0.2 & 0.45 \end{pmatrix}\). A is absorbing.

\(\begin{pmatrix} 0 & 0.5 & 0 & 0.35 \\ 0.4 & 0.2 & 0.15 & 0 \\ 0.6 & 0.3 & 0.35 & 0.25 \\ 0 & 0 & 0.5 & 0.4 \end{pmatrix}\)

\(\begin{pmatrix} 0.65 & 0.15 & 0.2 \\ 0.25 & 0.55 & 0.5 \\ 0.1 & 0.3 & 0.3 \end{pmatrix}\)

1. (a) Column 1 adds to 1.11, not 1
2. (b) a = 0.44, b = 0.37, c = 0.3

1. (a) \(\begin{pmatrix} 0.88 & 0.17 & 0 \\ 0.12 & 0.49 & 0 \\ 0 & 0.34 & 1 \end{pmatrix}\)
2. (b) Loops: H 0.88, I 0.49, D 1. Arrows: H to I 0.12, I to H 0.17, I to D 0.34.
3. (c) D: column D is (0, 0, 1)