Prob & Stats - Markov Chains (15 of 38) How to Find a Stable 3x3 Matrix By Michel van Biezen

Summary:

1. Introduction:

   • Previous videos covered finding the stable distribution matrix with a 2x2 probability matrix and states.
   • This video extends the example to a 3x3 scenario with three states: A, B, and C.
   • Objective: Determine the final distribution for these three states using Markov chains.

2. Probability Matrix Setup:

   • Explaining the concept of the probability matrix and its role in Markov chains.
   • Constructing the 3x3 probability matrix based on the given transition probabilities.

3. Equation Establishment:

   • Formulating equations by multiplying the probability matrix with the stable matrix.
   • Ensuring that the sum of each column equals 1 to maintain probability constraints.

4. Equation Solving:

   • Using algebraic manipulation to solve equations for the stable distribution of A, B, and C.
   • Expressing A, B, and C in terms of each other to find their final proportions.

5. Solving for A:

   • Setting up equations to solve for A in terms of B and C.
   • Simplifying the equation to find the proportion of A in the stable matrix.

6. Solving for B:

   • Replacing A with its value in terms of B and C.
   • Performing calculations to find the proportion of B in the stable matrix.

7. Solving for C:

   • Expressing C in terms of B and substituting the known value of B.
   • Calculating the proportion of C in the stable matrix.

8. Finalizing the Stable Distribution Matrix:

   • Summarizing the proportions of A, B, and C in the stable matrix.
   • Ensuring that the proportions sum up to 1, indicating a valid distribution.

9. Conclusion:

   • Reinforcing the concept of the stable distribution matrix in Markov chains.
   • Demonstrating the algebraic approach to finding stable distributions, applicable to larger matrices.
   • Emphasizing the importance of computational tools while acknowledging the algebraic method's viability.