Transition matrices and diagrams Problem set A Answers

1. (a) \(\begin{pmatrix} 0.64 & 0.27 \\ 0.36 & 0.73 \end{pmatrix}\)
2. (b) \(\begin{pmatrix} 0.12 & 0.59 \\ 0.88 & 0.41 \end{pmatrix}\)

\(\begin{pmatrix} 0.52 & 0.23 & 0.14 \\ 0.31 & 0.46 & 0.19 \\ 0.17 & 0.31 & 0.67 \end{pmatrix}\)

\(\begin{pmatrix} 0.31 & 0.12 & 0.28 & 0.22 \\ 0.27 & 0.36 & 0.07 & 0.29 \\ 0.18 & 0.33 & 0.46 & 0.38 \\ 0.24 & 0.19 & 0.19 & 0.11 \end{pmatrix}\)

1. (a) Loops: A 0.38, B 0.43. Arrows: A to B 0.62, B to A 0.57.
2. (b) Loops: A 0.66, B 0.52, C 0.23. Arrows: A to B 0.12, A to C 0.22, B to C 0.48, C to A 0.41, C to B 0.36.

\(\begin{pmatrix} 0.62 & 0.27 \\ 0.38 & 0.73 \end{pmatrix}\)

\(\begin{pmatrix} 0.55 & 0.4 & 0 \\ 0.3 & 0.4 & 0 \\ 0.15 & 0.2 & 1 \end{pmatrix}\). C is absorbing.

\(\begin{pmatrix} 0.3 & 0.25 & 0.1 & 0 \\ 0.5 & 0 & 0.2 & 0 \\ 0 & 0.75 & 0.4 & 0.6 \\ 0.2 & 0 & 0.3 & 0.4 \end{pmatrix}\)

\(\begin{pmatrix} 0.8 & 0.1 & 0.15 \\ 0.12 & 0.7 & 0.25 \\ 0.08 & 0.2 & 0.6 \end{pmatrix}\)

1. (a) Column 1 adds to 0.91, not 1
2. (b) a = 0.18, b = 0.33, c = 0.45

1. (a) \(\begin{pmatrix} 0.72 & 0.08 & 0 \\ 0.28 & 0.55 & 0 \\ 0 & 0.37 & 1 \end{pmatrix}\)
2. (b) Loops: C 0.72, L 0.55, D 1. Arrows: C to L 0.28, L to C 0.08, L to D 0.37.
3. (c) D: column D is (0, 0, 1)