4.19 Transition matrices and Markov chains

4.19 Transition matrices and Markov chains Quiz

IB Maths: Applications and Interpretation HL: Quiz

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  1. 1. Which matrix could be a transition matrix with columns as 'from' states?

    • A. (0.50.20.40.8)\begin{pmatrix} 0.5 & 0.2 \\ 0.4 & 0.8 \end{pmatrix}
    • B. (0.50.20.50.8)\begin{pmatrix} 0.5 & 0.2 \\ 0.5 & 0.8 \end{pmatrix}
    • C. (0.50.50.20.8)\begin{pmatrix} 0.5 & 0.5 \\ 0.2 & 0.8 \end{pmatrix}
    • D. (0.60.20.50.8)\begin{pmatrix} 0.6 & 0.2 \\ 0.5 & 0.8 \end{pmatrix}

  2. 2. T=(0.90.50.10.5)T=\begin{pmatrix} 0.9 & 0.5 \\ 0.1 & 0.5 \end{pmatrix} and s0=(10)\mathbf{s}_0=\begin{pmatrix} 1 \\ 0 \end{pmatrix}. Find s1\mathbf{s}_1.

    • A. (0.50.5)\begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}
    • B. (0.10.9)\begin{pmatrix} 0.1 \\ 0.9 \end{pmatrix}
    • C. (10)\begin{pmatrix} 1 \\ 0 \end{pmatrix}
    • D. (0.90.1)\begin{pmatrix} 0.9 \\ 0.1 \end{pmatrix}

  3. 3. For T=(0.90.50.10.5)T=\begin{pmatrix} 0.9 & 0.5 \\ 0.1 & 0.5 \end{pmatrix}, what is the long-term probability of the first state?

    • A. 56\frac56
    • B. 0.90.9
    • C. 16\frac16
    • D. 0.50.5

  4. 4. Which statement about the eigenvalue 11 of a transition matrix is correct?

    • A. It never occurs.
    • B. It gives the probability of the first state.
    • C. Its eigenvector, scaled to sum to 1, is the steady state.
    • D. It exists only for 3×33\times3 matrices.

  5. 5. A 2×22\times2 transition matrix TT and initial state s0\mathbf{s}_0 are given. What does T4s0T^4\mathbf{s}_0 give?

    • A. The steady state exactly.
    • B. The state matrix after 3 transitions.
    • C. The state matrix after 4 transitions.
    • D. The initial state four steps earlier.

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