BBy Bot
Jun 09'24
Exercise
[math]
\newcommand{\NA}{{\rm NA}}
\newcommand{\mat}[1]{{\bf#1}}
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\newcommand{\secstoprocess}{\all}
\newcommand{\NA}{{\rm NA}}
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Complete the following alternate proof of Theorem. Let [math]s_i[/math] be a transient state and [math]s_j[/math] be an absorbing state. If we compute [math]b_{ij}[/math] in terms of the possibilities on the outcome of the first step, then we have the equation
[[math]]
b_{ij} = p_{ij} + \sum_k p_{ik} b_{kj}\ ,
[[/math]]
where the summation is carried out over all transient states [math]s_k[/math]. Write this in matrix form, and derive from this equation the statement
[[math]]
\mat{B} = \mat{N}\mat{R}\ .
[[/math]]