Question

12. For a Markov chain {X_n, n ? 0} with transition probabilities P_{i,j}, consider the conditional probability that X_n = m given that the chain started at time 0 in state i and has not yet entered state r by time n, where r is a specified state not equal to either i or m. We are interested in whether this conditional probability is equal to the n stage transition probability of a Markov chain whose state space does not include state r and whose transition probabilities are Q_{i,j} = frac{P_{i,j}}{1 - P_{i,r}}, i, j ? r Either prove the equality P{X_n = m|X_0 = i, X_k ? r, k = 1, ..., n} = Q^n_{i,m} or construct a counterexample.

          12. For a Markov chain {X_n, n ? 0} with transition probabilities P_{i,j}, consider the conditional probability that X_n = m given that the chain started at time 0 in state i and has not yet entered state r by time n, where r is a specified state not equal to either i or m. We are interested in whether this conditional probability is equal to the n stage transition probability of a Markov chain whose state space does not include state r and whose transition probabilities are

Q_{i,j} = frac{P_{i,j}}{1 - P_{i,r}}, i, j ? r

Either prove the equality

P{X_n = m|X_0 = i, X_k ? r, k = 1, ..., n} = Q^n_{i,m}

or construct a counterexample.

$12. For a Markov chain Xn, n ? 0 with transition probabilities Pi,j, consider the conditional probability that Xn = m given that the chain started at time 0 in state i and has not yet entered state r by time n, where r is a specified state not equal to either i or m. We are interested in whether this conditional probability is equal to the n stage transition probability of a Markov chain whose state space does not include state r and whose transition probabilities are Qi,j = fracPi,j1 - Pi,r, i, j ? r Either prove the equality PXn = m|X0 = i, Xk ? r, k = 1, ..., n = Q^ni,m or construct a counterexample.$

Added by Rebecca V.

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