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Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Understanding State Transition Diagrams

### Task: Write Down the Transition Matrix

Here, we have a state transition diagram featuring five states. Your task is to write down the transition matrix associated with this diagram. A transition matrix represents the probabilities of moving from one state to another in a matrix format.

### State Transition Diagram
The diagram consists of five states, labeled 1 through 5. The transitions between these states are marked with the respective probabilities. Each state can transition to one or more other states, and these transitions are indicated with arrows and labeled with probabilities. Here's a detailed breakdown of the state transitions:

- **State 1:**
  - Transitions to State 1 with probability \( \frac{1}{3} \)
  - Transitions to State 2 with probability \( \frac{2}{3} \)

- **State 2:**
  - Transitions to State 2 with probability \( \frac{1}{3} \)
  - Transitions to State 3 with probability \( \frac{1}{3} \)
  - Transitions to State 4 with probability \( \frac{1}{3} \)

- **State 3:**
  - Transitions to State 5 with probability \( \frac{1}{2} \)
  - Transitions to State 2 with probability \( \frac{1}{2} \)

- **State 4:**
  - Transitions to State 2 with probability \( \frac{2}{3} \)
  - Transitions to State 1 with probability \( \frac{1}{3} \)

- **State 5:**
  - Transitions to State 1 with probability \( \frac{1}{2} \)
  - Transitions to State 4 with probability \( \frac{1}{2} \)

Below this detailed transition diagram, you are prompted to write down the transition matrix.

### Transition Matrix
To write the transition matrix, list the probabilities of transitioning from each state (rows) to every other state (columns). 

Here is an example format for the transition matrix based on the provided diagram:

| From \ To | 1     | 2     | 3     | 4     | 5     |
|-----------|-------|-------|-------|-------|-------|
| **1**     | 1/3   | 2/3   | 0     |
Transcribed Image Text:--- ## Understanding State Transition Diagrams ### Task: Write Down the Transition Matrix Here, we have a state transition diagram featuring five states. Your task is to write down the transition matrix associated with this diagram. A transition matrix represents the probabilities of moving from one state to another in a matrix format. ### State Transition Diagram The diagram consists of five states, labeled 1 through 5. The transitions between these states are marked with the respective probabilities. Each state can transition to one or more other states, and these transitions are indicated with arrows and labeled with probabilities. Here's a detailed breakdown of the state transitions: - **State 1:** - Transitions to State 1 with probability \( \frac{1}{3} \) - Transitions to State 2 with probability \( \frac{2}{3} \) - **State 2:** - Transitions to State 2 with probability \( \frac{1}{3} \) - Transitions to State 3 with probability \( \frac{1}{3} \) - Transitions to State 4 with probability \( \frac{1}{3} \) - **State 3:** - Transitions to State 5 with probability \( \frac{1}{2} \) - Transitions to State 2 with probability \( \frac{1}{2} \) - **State 4:** - Transitions to State 2 with probability \( \frac{2}{3} \) - Transitions to State 1 with probability \( \frac{1}{3} \) - **State 5:** - Transitions to State 1 with probability \( \frac{1}{2} \) - Transitions to State 4 with probability \( \frac{1}{2} \) Below this detailed transition diagram, you are prompted to write down the transition matrix. ### Transition Matrix To write the transition matrix, list the probabilities of transitioning from each state (rows) to every other state (columns). Here is an example format for the transition matrix based on the provided diagram: | From \ To | 1 | 2 | 3 | 4 | 5 | |-----------|-------|-------|-------|-------|-------| | **1** | 1/3 | 2/3 | 0 |
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