Transition matrices and diagrams Problem set A

A Markov chain has two states, A and B. Write down the transition matrix in each case.

1. (a) \(P(A \to A) = 0.64\); \(P(B \to A) = 0.27\).
2. (b) \(P(A \to B) = 0.88\); \(P(B \to B) = 0.41\).

A Markov chain has three states, A, B and C. The known probabilities are: \(P(A \to A) = 0.52\); \(P(A \to B) = 0.31\); \(P(B \to B) = 0.46\); \(P(B \to C) = 0.31\); \(P(C \to A) = 0.14\); \(P(C \to C) = 0.67\). Write down the transition matrix.

A Markov chain has four states, A, B, C and D. The probabilities of changing state are: \(P(A \to B) = 0.27\); \(P(A \to C) = 0.18\); \(P(A \to D) = 0.24\); \(P(B \to A) = 0.12\); \(P(B \to C) = 0.33\); \(P(B \to D) = 0.19\); \(P(C \to A) = 0.28\); \(P(C \to B) = 0.07\); \(P(C \to D) = 0.19\); \(P(D \to A) = 0.22\); \(P(D \to B) = 0.29\); \(P(D \to C) = 0.38\). Find the transition matrix.

Draw the transition diagram for each matrix. The rows and columns are in the order of the states shown.

1. (a) States A, B: \(\mathbf{T} = \begin{pmatrix} 0.38 & 0.57 \\ 0.62 & 0.43 \end{pmatrix}\)
2. (b) States A, B, C: \(\mathbf{T} = \begin{pmatrix} 0.66 & 0 & 0.41 \\ 0.12 & 0.52 & 0.36 \\ 0.22 & 0.48 & 0.23 \end{pmatrix}\)

Write down the transition matrix for the transition diagram, with the states in the order A, B.

Write down the transition matrix for the transition diagram, with the states in the order A, B, C. Say which state is absorbing.

Write down the transition matrix for the transition diagram, with the states in the order A, B, C, D.

Three cafes, A, B and C, share the same street. Each week, 80% of the customers of cafe A return to A and 12% go to B. Of the customers of B, 70% return to B and 20% go to C. Of the customers of C, 15% go to A and 25% go to B. Write down the transition matrix for the weekly movement of customers, with the cafes in the order A, B, C.

Answer each part.

1. (a) Explain why \(\begin{pmatrix} 0.52 & 0.28 \\ 0.39 & 0.72 \end{pmatrix}\) cannot be a transition matrix.
2. (b) \(\mathbf{T} = \begin{pmatrix} 0.42 & a & 0.09 \\ b & 0.31 & 0.46 \\ 0.25 & 0.51 & c \end{pmatrix}\) is a transition matrix. Find \(a\), \(b\) and \(c\).

A phone battery is checked each morning. It is charged (C), low (L) or dead (D). If it is charged, it is still charged the next morning with probability 0.72, and low with probability 0.28. If it is low, it is charged the next morning with probability 0.08 and dead with probability 0.37. A dead battery is replaced only at the end of the term, so it stays dead.

1. (a) Write down the transition matrix, with the states in the order C, L, D.
2. (b) List the loops and arrows of the transition diagram.
3. (c) State the absorbing state and say how you can see it in the matrix.