Lesson 2

Let us denote the population of Michigan fans by \(x\) and the population of Michigan State fans by \(y\). The number of fans of each team remains constant from year to year, hence, \[\begin{cases}x = 0.97x + 0.05y\\ y = 0.03x + 0.95y\end{cases}\] Solving these equations gives \(x = \frac{5}{3}y\), which implies that Michigan fans constitute \(x = \frac{5}{8}\) of the total fan population.

In terms of Markov chains, this solution can be stated as follows. The transition matrix of the considered process is \[\mathbf{T}= \begin{bmatrix}0.97 & 0.05 \\ 0.03 & 0.95\end{bmatrix}\] Since the number of fans of each team remains constant over time, the distribution vector \((x,y)^{T}\) satisfies the equation \(\mathbf{T}(x,y)^{T} = (x,y)^{T}\). \[\mathbf{T}(x,y)^{T} = \begin{bmatrix}0.97 & 0.05 \\ 0.03 & 0.95\end{bmatrix} \cdot \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0.97x + 0.05y \\ 0.03x + 0.95y\end{bmatrix} = \begin{bmatrix}x \\ y\end{bmatrix}\] Solving these equations gives \(x = \frac{5}{3}y\), which implies that Michigan fans constitute \(x = \frac{5}{8}\) of the total fan population.