() If volume is high this week, then next week it will be high with a probability of 0.8 and low with a probability of 0.2. (ii) If volume is low this week then it will be high next week with a probability of 0.5. Assume that state 1 is high volume and that state 2 is low volume. (1) Find the transition matrix for this Markov process. 0.8 0.2 P = 0.5 0.5 (2) If the volume this week is high, what is the probability that the volume will be high two weeks from now? .74 (3) What is the probability that volume will be high for three consecutive weeks? 田
() If volume is high this week, then next week it will be high with a probability of 0.8 and low with a probability of 0.2. (ii) If volume is low this week then it will be high next week with a probability of 0.5. Assume that state 1 is high volume and that state 2 is low volume. (1) Find the transition matrix for this Markov process. 0.8 0.2 P = 0.5 0.5 (2) If the volume this week is high, what is the probability that the volume will be high two weeks from now? .74 (3) What is the probability that volume will be high for three consecutive weeks? 田
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Please solve and explain the solution to part (3) and show all steps :)
(3) What is the probability that the volume will be high for three consecutive weeks?
![### Understanding Markov Processes
In this section, we explore a Markov process involving weekly volume states. We define two states for this model:
- **State 1**: High Volume
- **State 2**: Low Volume
#### Problem Setup
We establish the following probabilities for the transitions:
(i) If the volume is high this week, the probability that it remains high next week is 0.8, and the probability that it becomes low is 0.2.
(ii) If the volume is low this week, the probability that it becomes high next week is 0.5, and the probability that it remains low is 0.5.
#### Transition Matrix
To represent this process, we set up the transition matrix \( P \):
\[
P = \begin{bmatrix}
0.8 & 0.2 \\
0.5 & 0.5
\end{bmatrix}
\]
Each entry \( P_{ij} \) in the matrix represents the probability of transitioning from state \( i \) to state \( j \).
#### Exercises
1. **Finding the Transition Matrix**: Using the given probabilities, the transition matrix is provided above.
2. **Probability of High Volume in Two Weeks**: Given the volume is high this week, we calculate the probability of it being high two weeks from now. The result is \( 0.74 \).
3. **Probability of High Volume for Three Consecutive Weeks**: Calculate the likelihood of having high volume over three continuous weeks. (The answer isn't provided; use the transition matrix to compute this.)
This educational content is designed to help you understand how to use transition matrices in Markov processes to predict future states based on current conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36a79c17-bf73-44c3-9503-d2f30f855206%2Faf86dab4-3167-4f03-a3a8-023d00c53bf4%2Fiwoue2t_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Markov Processes
In this section, we explore a Markov process involving weekly volume states. We define two states for this model:
- **State 1**: High Volume
- **State 2**: Low Volume
#### Problem Setup
We establish the following probabilities for the transitions:
(i) If the volume is high this week, the probability that it remains high next week is 0.8, and the probability that it becomes low is 0.2.
(ii) If the volume is low this week, the probability that it becomes high next week is 0.5, and the probability that it remains low is 0.5.
#### Transition Matrix
To represent this process, we set up the transition matrix \( P \):
\[
P = \begin{bmatrix}
0.8 & 0.2 \\
0.5 & 0.5
\end{bmatrix}
\]
Each entry \( P_{ij} \) in the matrix represents the probability of transitioning from state \( i \) to state \( j \).
#### Exercises
1. **Finding the Transition Matrix**: Using the given probabilities, the transition matrix is provided above.
2. **Probability of High Volume in Two Weeks**: Given the volume is high this week, we calculate the probability of it being high two weeks from now. The result is \( 0.74 \).
3. **Probability of High Volume for Three Consecutive Weeks**: Calculate the likelihood of having high volume over three continuous weeks. (The answer isn't provided; use the transition matrix to compute this.)
This educational content is designed to help you understand how to use transition matrices in Markov processes to predict future states based on current conditions.
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