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. The steady-state probabilities represent the long-term proportions of customers buying each newspaper in the city.
To find the eigenvector, we start by writing the transition matrix as:
M = [0.65 0.35; 0.45 0.55]
Next, we need to solve the equation (M - I)v = 0, where v is the eigenvector we are looking for and I is the identity matrix. This equation represents finding the null space of (M - I).
Let's set up the equation:
[M - I]v = 0
Substituting the values, we get:
[0.65 - 1 0.35; 0.45 0.55 - 1]v = 0
Simplifying further:
[-0.35 0.35; 0.45 -0.45]v = 0
Next, we solve this system of linear equations to find the eigenvector:
-0.35v1 + 0.35v2 = 0
0.45v1 - 0.45v2 = 0
To solve the equations, we can set v1 = t (a parameter) and express v2 in terms of t:
v2 = v1
Now, we can normalize the eigenvector by setting the sum of its components equal to 1:
t + t = 1
2t = 1
t = 1/2
Therefore, the eigenvector is (1/2, 1/2).
The steady-state probabilities represent the long-term proportions of customers buying each newspaper in the city. In this case, the steady-state probabilities are (1/2, 1/2), which means that in the long run, half of the customers will buy the Tribune and half will buy the Daily News. This indicates a balanced market share between the two newspapers.