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| \(i\in C(J)\) | | \(j\in C(J)\) |
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\(f_{ij}=1\)
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| \(i\in C(J)\) | | \(j\notin C(J)\) |
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\(f_{ij}=0\)
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| \(i\in T\) | | \(j\) recurrent but not absorbent, hence in a closed set with other states |
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| Use formula in page 5.5 lecture notes \(f_{ij}={\displaystyle \sum \limits _{k\in T}} p_{ik}f_{kj}+{\displaystyle \sum \limits _{k\in C\left ( J\right ) }} p_{ik}\) \(F=\left ( I-Q\right ) ^{-1}Z\) , hence just needs to find \(Z\) |
| \(z_{ij}=\)probability that transient state \(i\) will reach class that contains \(j\) |
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| \(i\in T\) | | \(j\) is an absorbent |
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\(f_{ij}=\left [ \left ( I-Q\right ) ^{-1}R\right ] _{i,j}\)
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We know eventually \(p_{ij}=0\) for \(i,j\in Q\), but can we talk about \(f_{ij}\) here?
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