Steady state and long-term probabilities Problem set A

\(\mathbf{T} = \begin{pmatrix} 0.67 & 0.14 & 0.19 \\ 0.21 & 0.79 & 0.28 \\ 0.12 & 0.07 & 0.53 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.6 \\ 0.1 \\ 0.3 \end{pmatrix}\). Find \(\mathbf{T}^{50} \mathbf{s}_0\) and \(\mathbf{T}^{100} \mathbf{s}_0\). Use them to write down the long-term state vector.

\(\mathbf{T} = \begin{pmatrix} 0.52 & 0.15 & 0.22 \\ 0.29 & 0.57 & 0.31 \\ 0.19 & 0.28 & 0.47 \end{pmatrix}\). The steady-state vector is \(\mathbf{s} = (s_1, s_2, s_3)\). Use \(\mathbf{T}\mathbf{s} = \mathbf{s}\) to write down a system of three linear equations that can be solved for \(s_1, s_2, s_3\). Do not solve it.

Find the exact steady-state vector by setting up and solving a system of equations.

1. (a) \(\mathbf{T} = \begin{pmatrix} 0.79 & 0.34 \\ 0.21 & 0.66 \end{pmatrix}\)
2. (b) \(\mathbf{T} = \begin{pmatrix} 0.58 & 0.28 & 0.2 \\ 0.23 & 0.27 & 0.22 \\ 0.19 & 0.45 & 0.58 \end{pmatrix}\)

Find the exact steady-state vector for \(\mathbf{T} = \begin{pmatrix} 0.06 & 0.34 & 0.5 & 0.4 \\ 0.08 & 0.22 & 0.04 & 0.12 \\ 0.44 & 0.04 & 0.36 & 0.22 \\ 0.42 & 0.4 & 0.1 & 0.26 \end{pmatrix}\).

Use your calculator to find the steady-state vector. Give the entries to 3 significant figures.

1. (a) \(\mathbf{T} = \begin{pmatrix} 0.74 & 0.07 & 0.09 \\ 0.14 & 0.87 & 0.13 \\ 0.12 & 0.06 & 0.78 \end{pmatrix}\)
2. (b) \(\mathbf{T} = \begin{pmatrix} 0.38 & 0.04 & 0.22 & 0.04 \\ 0.16 & 0.02 & 0.1 & 0.5 \\ 0.4 & 0.44 & 0.48 & 0.2 \\ 0.06 & 0.5 & 0.2 & 0.26 \end{pmatrix}\)

\(\mathbf{T} = \begin{pmatrix} 0.12 & 0.16 & 0.22 & 0.06 \\ 0.5 & 0.12 & 0.46 & 0.44 \\ 0.28 & 0.38 & 0.02 & 0.04 \\ 0.1 & 0.34 & 0.3 & 0.46 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.25 \\ 0.25 \\ 0.4 \\ 0.1 \end{pmatrix}\). Find \(\mathbf{s}_5\).

The transition matrix \(\mathbf{T} = \begin{pmatrix} 0.72 & 0.19 \\ 0.28 & 0.81 \end{pmatrix}\) gives the probability that a customer's next phone is made by Aero (A) or by Brio (B), with the brands in the order A, B. Write down the probability that a customer who now has a Brio phone buys an Aero phone next.

Three types of land use, 1, 2 and 3, change each decade with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.77 & 0.11 & 0.09 \\ 0.16 & 0.74 & 0.21 \\ 0.07 & 0.15 & 0.7 \end{pmatrix}\), in that order.

1. (a) Find the long-term probability matrix, with entries to 3 significant figures.
2. (b) Hence state the long-term percentage of type 2.

A hockey team wins (W) or does not win (N) each match. If it wins a match, the probability that it wins the next is 0.68. If it does not win a match, the probability that it wins the next is 0.37.

1. (a) Write down the transition matrix.
2. (b) The team did not win its last match. Find the probability that it wins its third match after that.
3. (c) Find the exact steady-state vector, and state the long-term probability that the team wins a match.

\(\mathbf{T} = \begin{pmatrix} 0.74 & p \\ 0.26 & 1 − p \end{pmatrix}\) is a transition matrix with steady-state vector \(\mathbf{s} = \begin{pmatrix} 0.6 \\ 0.4 \end{pmatrix}\).

1. (a) Use \(\mathbf{T}\mathbf{s} = \mathbf{s}\) to write down an equation for p.
2. (b) Solve it to find p.