ex6 pg422-423
In the Land of Oz example(Example 11.1), change the transition matrix by making R an obsorting state. This gives
\[P = \begin{array} \s & R & N & S \\ R & 1 & 0 & 0 \\ N & 1/2 & 0 & 1/2 \\ S & 1/4 & 1/4 & 1/2 \end{array}\]
Find the fundamental matrix N, and also Nc and NR. Interpret the results.
the canonical form of P is:
\[P = \begin{array} \s & N & S & R \\ N & 0 & 1/2 & 1/2 \\ S & 1/4 & 1/2 & 1/4 \\ R & 0&0&1\end{array}\]
Therefore,
\[Q = \begin{bmatrix}0&1/2\\1/4&1/2 \end{bmatrix}\]
\[N = (I - Q)^{-1} \\= \begin{pmatrix} \begin{bmatrix} 1&0\\0&1 \end{bmatrix} - \begin{bmatrix}0&1/2\\1/4&1/2 \end{bmatrix} \end{pmatrix}^{-1} \\= \begin{pmatrix} \begin{bmatrix}1&-1/2\\-1/4&1/2 \end{bmatrix} \end{pmatrix}^{-1} \\ =\begin{bmatrix} 4/3&4/3 \\ 2/3&8/3 \end{bmatrix}\]
then,
\[Nc = \begin{bmatrix} 4/3&4/3 \\ 2/3&8/3 \end{bmatrix} \begin{bmatrix} 1\\1 \end{bmatrix} \\ = \begin{bmatrix} 8/3 \\ 10/3 \end{bmatrix}\] Thus, starting in nice weather or snow weather, the expected times to absorption are 8/3 and 10/3, respectively.
Finally,
\[R = \begin{bmatrix} 1/2 \\1/4 \end{bmatrix}\]
\[NR = \begin{bmatrix} 4/3&4/3 \\ 2/3&8/3 \end{bmatrix} \begin{bmatrix} 1/2 \\1/4 \end{bmatrix} \\= \begin{bmatrix} 1\\1 \end{bmatrix}\]
Here, it tells that either starting from nice day or snow day, there is probability 1 of absorption in rainny day.