(4) If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and finally state 1 again)?
(4) If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and finally state 1 again)?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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I need help with the last one (#4)
![# Markov Chain Transition Matrix Analysis
## Transition Matrix
A Markov chain has the following transition matrix:
\[
P = \begin{bmatrix}
0.6 & 0.4 \\
0.8 & 0.2
\end{bmatrix}
\]
(Note: For questions 1, 2, and 4, express your answers as decimal fractions rounded to 4 decimal places if they have more than 4 decimal places.)
---
### Questions
1. **If, on the first observation, the system is in state 1, what is the probability that it is in state 1 on the second observation?**
**Answer:** 0.6
2. **If, on the first observation, the system is in state 1, what is the probability that it is in state 1 on the third observation?**
**Answer:** 0.68
3. **If, on the first observation, the system is in state 2, what state is the system most likely to occupy on the third observation? (If there is more than one such state, which is the first one.)**
**Answer:** 1
4. **If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and finally state 1 again)?**
**Answer:** 0.128
---
### Explanation of Concepts
A **Markov chain** is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. The **transition matrix \( P \)** defines the probabilities of moving from one state to another in a single step. Each element in the matrix represents the probability of transitioning from state \( i \) to state \( j \).
In this case, the matrix shows how likely it is to move between two states (state 1 and state 2) with the given probabilities for each possible transition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4da1f4c8-4970-49e9-beef-0939f39f632e%2F527d3740-4f22-48c1-980c-470c2466e5de%2Fgbpfwq_processed.png&w=3840&q=75)
Transcribed Image Text:# Markov Chain Transition Matrix Analysis
## Transition Matrix
A Markov chain has the following transition matrix:
\[
P = \begin{bmatrix}
0.6 & 0.4 \\
0.8 & 0.2
\end{bmatrix}
\]
(Note: For questions 1, 2, and 4, express your answers as decimal fractions rounded to 4 decimal places if they have more than 4 decimal places.)
---
### Questions
1. **If, on the first observation, the system is in state 1, what is the probability that it is in state 1 on the second observation?**
**Answer:** 0.6
2. **If, on the first observation, the system is in state 1, what is the probability that it is in state 1 on the third observation?**
**Answer:** 0.68
3. **If, on the first observation, the system is in state 2, what state is the system most likely to occupy on the third observation? (If there is more than one such state, which is the first one.)**
**Answer:** 1
4. **If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and finally state 1 again)?**
**Answer:** 0.128
---
### Explanation of Concepts
A **Markov chain** is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. The **transition matrix \( P \)** defines the probabilities of moving from one state to another in a single step. Each element in the matrix represents the probability of transitioning from state \( i \) to state \( j \).
In this case, the matrix shows how likely it is to move between two states (state 1 and state 2) with the given probabilities for each possible transition.
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