Question

A Markov chain on the state space $S = \{1, 2, 3, 4\}$ has the transition probability matrix $P = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1/3 & 0 & 2/3 & 0 \\ 0 & 1/2 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \end{pmatrix}$ 1. Write the matrix in the canonical form if applicable. 2. Find the probability that the absorption occurs at state 1 starting from state 2. 3. Determine the mean time to reach state 1 or 4 (the absorbing states) starting from state 2 4. Determine the mean time to reach state 4 starting from state 1 5. Determine the mean time spent in state 3 given that we start from state 2 before being absorbed into either state 1 or state 4.

Name: a markov chain on the state space s 1 2 3 4 has the transition probability matrix p beginpmatrix 1 0 0 0 13 0 23 0 0 12 0 12 0 0 0 1 endpmatrix 1 write the matrix in the canonical form if ap 00013
Uploaded: 2023-03-16T04:15:14-08:00
Duration: 5 min 49 s
Channel: Sri K
Description: a markov chain on the state space s 1 2 3 4 has the transition probability matrix p beginpmatrix 1 0 0 0 13 0 23 0 0 12 0 12 0 0 0 1 endpmatrix 1 write the matrix in the canonical form if ap 00013

          A Markov chain on the state space $S = \{1, 2, 3, 4\}$ has the transition probability matrix
$P = \begin{pmatrix}
1 & 0 & 0 & 0 \\
1/3 & 0 & 2/3 & 0 \\
0 & 1/2 & 0 & 1/2 \\
0 & 0 & 0 & 1
\end{pmatrix}$
1. Write the matrix in the canonical form if applicable.
2. Find the probability that the absorption occurs at state 1 starting from state 2.
3. Determine the mean time to reach state 1 or 4 (the absorbing states) starting from state 2
4. Determine the mean time to reach state 4 starting from state 1
5. Determine the mean time spent in state 3 given that we start from state 2 before being absorbed into either state 1 or state 4.

a markov chain on the state space s 1 2 3 4 has the transition probability matrix p beginpmatrix 1 0 0 0 13 0 23 0 0 12 0 12 0 0 0 1 endpmatrix 1 write the matrix in the canonical form if ap 00013

Added by Christopher O.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

03/16/2023

Step 1

The canonical form for an absorbing Markov chain is: $P = \begin{pmatrix} I & 0 \\ R & Q \end{pmatrix}$ where I is the identity matrix, 0 is a zero matrix, R is the matrix of probabilities of going from a transient state to an absorbing state, and Q is the matrix Show more…

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A Markov chain on the state space $S = \{1, 2, 3, 4\}$ has the transition probability matrix $P = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1/3 & 0 & 2/3 & 0 \\ 0 & 1/2 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \end{pmatrix}$ 1. Write the matrix in the canonical form if applicable. 2. Find the probability that the absorption occurs at state 1 starting from state 2. 3. Determine the mean time to reach state 1 or 4 (the absorbing states) starting from state 2 4. Determine the mean time to reach state 4 starting from state 1 5. Determine the mean time spent in state 3 given that we start from state 2 before being absorbed into either state 1 or state 4.