(1) Find the two-step transition matrix .18 .7 .12 P(2) = 1 .24 .42 .34 ... .
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Transition Matrices for Markov Chains
#### (1) Two-Step Transition Matrix
The two-step transition matrix, \( P(2) \), is defined as follows:
\[
P(2) =
\begin{bmatrix}
0.18 & 0.7 & 0.12 \\
0 & 1 & 0 \\
0.24 & 0.42 & 0.34 \\
\end{bmatrix}
\]
This matrix represents the probability of moving from one state to another in two steps. Each element \( p_{ij} \) represents the probability of transitioning from state \( i \) to state \( j \) in two steps.
#### (2) Three-Step Transition Matrix
The three-step transition matrix, \( P(3) \), is shown as a placeholder matrix:
\[
P(3) =
\begin{bmatrix}
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
\end{bmatrix}
\]
You need to calculate \( P(3) \) by multiplying the one-step transition matrix by itself three times.
#### (3) Four-Step Transition Matrix
The four-step transition matrix, \( P(4) \), is also shown as a placeholder matrix:
\[
P(4) =
\begin{bmatrix}
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
\end{bmatrix}
\]
This requires calculating by multiplying the one-step transition matrix by itself four times to determine the probability transitions over four steps.
In summary, the transition matrices provide a way to model different step transitions within a Markov chain, from a given state to another or the same state across multiple iterations, which is crucial in understanding long-term state behaviors in stochastic processes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac73edbf-297f-4aed-9f9e-4c41e8d7aed8%2Fe596df00-981f-4353-8e13-1beeeca9f04b%2Favw6pa_processed.png&w=3840&q=75)
Transcribed Image Text:### Transition Matrices for Markov Chains
#### (1) Two-Step Transition Matrix
The two-step transition matrix, \( P(2) \), is defined as follows:
\[
P(2) =
\begin{bmatrix}
0.18 & 0.7 & 0.12 \\
0 & 1 & 0 \\
0.24 & 0.42 & 0.34 \\
\end{bmatrix}
\]
This matrix represents the probability of moving from one state to another in two steps. Each element \( p_{ij} \) represents the probability of transitioning from state \( i \) to state \( j \) in two steps.
#### (2) Three-Step Transition Matrix
The three-step transition matrix, \( P(3) \), is shown as a placeholder matrix:
\[
P(3) =
\begin{bmatrix}
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
\end{bmatrix}
\]
You need to calculate \( P(3) \) by multiplying the one-step transition matrix by itself three times.
#### (3) Four-Step Transition Matrix
The four-step transition matrix, \( P(4) \), is also shown as a placeholder matrix:
\[
P(4) =
\begin{bmatrix}
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
[ ] & [ ] & [ ] \\
\end{bmatrix}
\]
This requires calculating by multiplying the one-step transition matrix by itself four times to determine the probability transitions over four steps.
In summary, the transition matrices provide a way to model different step transitions within a Markov chain, from a given state to another or the same state across multiple iterations, which is crucial in understanding long-term state behaviors in stochastic processes.
![(4) Using the transition matrices you computed above, find \( p_{22}(2) \), \( p_{22}(3) \), and \( p_{22}(4) \):
\[
p_{22}(2) = \boxed{\phantom{}}.
\]
\[
p_{22}(3) = \boxed{\phantom{}}.
\]
\[
p_{22}(4) = \boxed{\phantom{}}.
\]
(5) What do you think \( p_{22}(20) \) is?
\[
p_{22}(20) = \boxed{\phantom{}}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac73edbf-297f-4aed-9f9e-4c41e8d7aed8%2Fe596df00-981f-4353-8e13-1beeeca9f04b%2Fvhcuhek_processed.png&w=3840&q=75)
Transcribed Image Text:(4) Using the transition matrices you computed above, find \( p_{22}(2) \), \( p_{22}(3) \), and \( p_{22}(4) \):
\[
p_{22}(2) = \boxed{\phantom{}}.
\]
\[
p_{22}(3) = \boxed{\phantom{}}.
\]
\[
p_{22}(4) = \boxed{\phantom{}}.
\]
(5) What do you think \( p_{22}(20) \) is?
\[
p_{22}(20) = \boxed{\phantom{}}.
\]
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