Math Problem Statement

Find the equilibrium vector for the transition matrix.

left bracket Start 3 By 3 Matrix 1st Row 1st Column 0.75 2nd Column 0.10 3rd Column 0.15 2nd Row 1st Column 0.10 2nd Column 0.70 3rd Column 0.20 3rd Row 1st Column 0.10 2nd Column 0.35 3rd Column 0.55 EndMatrix right bracket

0.75

0.10

0.15

0.10

0.70

0.20

0.10

0.35

0.55

Question content area bottom

Part 1

The equilibrium vector is

enter your response here.

​(Type an integer or simplified fraction for each matrix​ element.)

Solution

To find the equilibrium vector for the given transition matrix, we solve the equation:

P⋅v⃗=v⃗P \cdot \vec{v} = \vec{v}

where PP is the transition matrix and v⃗=[v1,v2,v3]T\vec{v} = [v_1, v_2, v_3]^T is the equilibrium vector. This implies that the equilibrium vector remains unchanged after applying the transition matrix.

  1. Transition matrix:
0.75 & 0.10 & 0.15 \\ 0.10 & 0.70 & 0.20 \\ 0.10 & 0.35 & 0.55 \end{bmatrix}$$ 2. **Equations for equilibrium vector:** The equations from $$P \cdot \vec{v} = \vec{v}$$ are: $$\begin{aligned} 0.75v_1 + 0.10v_2 + 0.15v_3 &= v_1 \\ 0.10v_1 + 0.70v_2 + 0.20v_3 &= v_2 \\ 0.10v_1 + 0.35v_2 + 0.55v_3 &= v_3 \end{aligned}$$ We rewrite them as: $$\begin{aligned} -0.25v_1 + 0.10v_2 + 0.15v_3 &= 0 \\ 0.10v_1 - 0.30v_2 + 0.20v_3 &= 0 \\ 0.10v_1 + 0.35v_2 - 0.45v_3 &= 0 \end{aligned}$$ Additionally, we enforce the condition that the probabilities sum to 1: $$v_1 + v_2 + v_3 = 1$$ 3. **Solving the system of equations:** I will solve this system to find the equilibrium vector. numpy as np # Transition matrix P = np.array([ [0.75, 0.10, 0.15], [0.10, 0.70, 0.20], [0.10, 0.35, 0.55] ]) # Subtract identity matrix to set up P*v = v => (P - I)*v = 0 A = P - np.eye(3) # Add the condition that v1 + v2 + v3 = 1 A = np.vstack([A.T, np.ones(3)]) b = np.array([0, 0, 0, 1]) # Solve for v v = np.linalg.lstsq(A, b, rcond=None)[0] vThe equilibrium vector is: $$\vec{v} = \begin{bmatrix} \frac{2}{7} \\ \frac{3}{7} \\ \frac{2}{7} \end{bmatrix}$$ This means the probabilities stabilize to $$\frac{2}{7}$$ for state 1, $$\frac{3}{7}$$ for state 2, and $$\frac{2}{7}$$ for state 3. ### Do you want more details or have any questions? ### Relative questions: 1. How is the equilibrium vector derived from the system of equations? 2. Why does the equilibrium vector represent a stable state in Markov processes? 3. How would changes to the transition matrix affect the equilibrium vector? 4. What does the sum-to-1 condition signify in terms of probabilities? 5. Can this method be extended to larger transition matrices? ### Tip: When working with Markov chains, always verify that the rows of the transition matrix sum to 1 to ensure it represents valid probabilities.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Markov Chains
Matrix Operations

Formulas

P * v = v
v1 + v2 + v3 = 1
(P - I) * v = 0

Theorems

Steady-State Theorem for Markov Chains

Suitable Grade Level

Undergraduate (Linear Algebra or Probability)

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