A city is served by two newspapers—the Tribune and the Daily News. Each Sunday reader purchases
one of the newspapers at a stand. The following transition matrix contains the probabilities of a
customer’s buying a particular newspaper in a week, given the newspaper purchased the previous
Sunday:
This Sunday Next Sunday
Tribune Daily News
Tribune 0.65 0.35
0.45 0.55
Determine the steady-state probabilities for this transition matrix algebraically and explain
what they mean.
To determine the steady-state probabilities for this transition matrix algebraically, we need to find the eigenvector associated with the eigenvalue of 1.
1. Write the transition matrix in equation form: P * X = X, where P is the transition matrix and X is the eigenvector.
2. Subtract the identity matrix from the transition matrix: P - I = 0, where I is the identity matrix.
3. Solve the equation (P - I) * X = 0 to find the eigenvector X.
Using the given transition matrix:
0.65 0.35
0.45 0.55
Subtracting the identity matrix:
(0.65 - 1) 0.35
0.45 (0.55 - 1)
Simplifying:
-0.35 0.35
0.45 -0.45
Now, we need to find the eigenvector X that satisfies (P - I) * X = 0:
-0.35X + 0.35X = 0
0.45X - 0.45X = 0
This system of equations has infinitely many solutions because it is a homogeneous system. Therefore, we can conclude that the steady-state probabilities for this transition matrix do not exist.
This means that there is no long-term equilibrium in newspaper purchases in this city. The probabilities of buying the Tribune or the Daily News do not converge to specific values, and the newspaper readership will continue to change over time without settling into a stable pattern.