a city is served by two newspapers-the Tribune and the Daily News. Each Sunday a reader purchases one of the newspapers at a stand. The following transition matrix contains the probabilities of a customers buying a particular newspaper in a week given the newspaper purchased the previous Sunday.

Question

a city is served by two newspapers-the Tribune and the Daily News. Each Sunday a reader purchases one of the newspapers at a stand. The following transition matrix contains the probabilities of a customers buying a particular newspaper in a week given the newspaper purchased the previous Sunday.

.65 .35
.45 .55
Determine the steady-state probabilities for this transition matrix algebraically and explain what they mean

Answer 1

To determine the steady-state probabilities for this transition matrix algebraically, we need to solve the equation πT = πT P, where πT is the row vector of steady-state probabilities and P is the transition matrix.

In this case, the transition matrix is:
P = [[0.65, 0.35],
[0.45, 0.55]]

Let's represent the steady-state probabilities as π = [x, y]. Rewriting the equation, we have:
[x, y] = [x, y] * [[0.65, 0.35],
[0.45, 0.55]]

Simplifying the equation, we get:
x = x * 0.65 + y * 0.45
y = x * 0.35 + y * 0.55

Now we can solve these two equations simultaneously to find the steady-state probabilities x and y.

From the first equation, we can rearrange it as x - 0.65x = 0.45y, which simplifies to:
0.35x = 0.45y
7x = 9y

Substituting the second equation into this, we get:
7x = 9(x * 0.35 + y * 0.55)
7x = 3.15x + 4.95y

Rearranging this equation, we have:
3.85x = 4.95y

Now, we can set up the equation to solve for one of the variables in terms of the other:
x = (4.95/3.85)y

Since these probabilities must add up to 1, we also have the equation:
x + y = 1

Substituting in the value of x, we get:
(4.95/3.85)y + y = 1
(4.95 + 3.85)/3.85)y = 1
(8.8/3.85)y = 1
y = (3.85/8.8)

Substituting this value back into the equation for x, we have:
x = (4.95/3.85)(3.85/8.8)
x = (4.95/8.8)

Therefore, the steady-state probabilities for this transition matrix are approximately:
x ≈ 0.56 (or 56%)
y ≈ 0.44 (or 44%)

These probabilities represent the long-term proportion of customers who will choose each newspaper on any given Sunday. In this case, approximately 56% of customers will choose the Tribune newspaper and 44% will choose the Daily News in the steady-state.