Question 10.5.3: Consider the unbiased random walk on {1, 2, 3, 4, 5} with ab...

Linear Algebra and Its Applications

Consider the unbiased random walk on {1, 2, 3, 4, 5} with absorbing boundaries studied in Example 1. Find the probability that the chain is absorbed into state 1 given that the chain starts at state 4.

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Placing the states in the order {1, 2, 3, 4, 5} gives the canonical form of the transition matrix:

\text { The matrix } Q \text { and the fundamental matrix } M=(I-Q)^{-1} \text { are }

so

A=S M=\left[\begin{array}{ccc}1 / 2 & 0 & 0 \\0 & 0 & 1 / 2\end{array}\right]\left[\begin{array}{ccc}3 / 2 & 1 & 1 / 2 \\1 & 2 & 1 \\1 / 2 & 1 & 3 / 2\end{array}\right]

The columns of A correspond to the transient states 2, 3, and 4 in that order, while the rows correspond to the absorbing states 1 and 5. The probability that the chain that started at state 4 is absorbed at state 1 is 1/4.

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