Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/markov-chains-transition-matrices/ahl-7-5/problems/

jmaths

IB Mathematics: Applications and Interpretation HL · AHL 7: Markov Chains & Transition Matrices · AHL 7.5

Markov Chains & Transition Matrices: exam review Exam review Answers

Section A: short questions

    1. (a) \(\mathbf{T}=\begin{pmatrix}0.5 & 0.25 & 0.4 \\ 0.3 & 0.6 & 0 \\ 0.2 & 0.15 & 0.6\end{pmatrix}\)
    2. (b) A diagram with vertices L, H, C and 8 arrows: loop at L: 0.5; L to H: 0.3; L to C: 0.2; H to L: 0.25; loop at H: 0.6; H to C: 0.15; C to L: 0.4; loop at C: 0.6
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.9 & 0.5 & 0 \\ 0.1 & 0.3 & 0 \\ 0 & 0.2 & 1\end{pmatrix}\)
    2. (b) D: its column is \((0, 0, 1)\), a 1 on the diagonal and 0 elsewhere
    3. (c) 0.044
    1. (a) 0.61
    2. (b) 64.2% attentive and 35.8% distracted
    1. (a) \(0.7s_1 + 0.2s_2 + 0.1s_3 = s_1\); \(0.2s_1 + 0.6s_2 + 0.3s_3 = s_2\); \(s_1+s_2+s_3=1\) (any two of the three row equations, with the sum)
    2. (b) \(\mathbf{s}=\begin{pmatrix}\frac{10}{29} \\ \frac{11}{29} \\ \frac{8}{29}\end{pmatrix}\)
    3. (c) 517
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.65 & 0.2 \\ 0.35 & 0.8\end{pmatrix}\)
    2. (b) 0.056
    3. (c) 0.507
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.8 & 0.35 \\ 0.2 & 0.65\end{pmatrix}\)
    2. (b) 383 attend, 217 miss
    3. (c) 382
    4. (d) \(\frac{7}{11}\)

Section B: long questions

    1. (a) \(\mathbf{T}=\begin{pmatrix}0.5 & 0.35 & 0.2 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.25 & 0.5\end{pmatrix}\)
    2. (b) 0.395
    3. (c) \(\begin{pmatrix}0.358 \\ 0.333 \\ 0.309\end{pmatrix}\)
    4. (d) \(\begin{pmatrix}0.357 & 0.357 & 0.357 \\ 0.333 & 0.333 & 0.333 \\ 0.31 & 0.31 & 0.31\end{pmatrix}\)
    5. (e) 35.7%
    6. (f) In the long run the chain settles to the steady state, which does not depend on the starting state
    1. (a) Squares 0 and 4. Once the chain is in an absorbing state it stays there
    2. (b) \(\mathbf{T}=\begin{pmatrix}1 & 0.45 & 0 & 0 & 0 \\ 0 & 0 & 0.45 & 0 & 0 \\ 0 & 0.55 & 0 & 0.45 & 0 \\ 0 & 0 & 0.55 & 0 & 0 \\ 0 & 0 & 0 & 0.55 & 1\end{pmatrix}\)
    3. (c) 0.272
    4. (d) 0.329
    5. (e) The long-term columns are different, because where the chain ends depends on where it starts
    6. (f) 66
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.85 & 0.1 \\ 0.15 & 0.9\end{pmatrix}\)
    2. (b) \(\lambda=1\) and \(\lambda=0.75\)
    3. (c) \(\lambda=1\): \(\begin{pmatrix}2 \\ 3\end{pmatrix}\); \(\lambda=0.75\): \(\begin{pmatrix}1 \\ -1\end{pmatrix}\) (or any multiples)
    4. (d) \(\mathbf{P}=\begin{pmatrix}2 & 1 \\ 3 & -1\end{pmatrix}\), \(\mathbf{D}=\begin{pmatrix}1 & 0 \\ 0 & 0.75\end{pmatrix}\)
    5. (e) \(0.4+0.2(0.75)^n\)
    6. (f) 0.4
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.45 & 0.3 & 0.15 & 0.1 \\ 0.2 & 0.45 & 0.2 & 0.3 \\ 0.1 & 0.25 & 0.3 & 0.2 \\ 0.25 & 0 & 0.35 & 0.4\end{pmatrix}\)
    2. (b) 0.236
    3. (c) \(\begin{pmatrix}265 \\ 296 \\ 207 \\ 232\end{pmatrix}\) (nearest whole numbers)
    4. (d) \(\mathbf{s}=\begin{pmatrix}\frac{545}{2084} \\ \frac{155}{521} \\ \frac{437}{2084} \\ \frac{241}{1042}\end{pmatrix}\)
    1. (a) \(\mathbf{T}=\begin{pmatrix}0 & 0.5 & 0.3 \\ 0.6 & 0.2 & 0.4 \\ 0.4 & 0.3 & 0.3\end{pmatrix}\)
    2. (b) \(\mathbf{T}\) has a zero entry but every entry of \(\mathbf{T}^{2}\) is positive, so it is regular. The smallest power is 2
    3. (c) eigenvector in proportion to (0.880, 1.160, 1.000); steady state (0.289, 0.382, 0.329)
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.4 & 0.2 & 0 & 0.5 \\ 0.6 & 0.3 & 0 & 0 \\ 0 & 0.5 & 0.6 & 0.1 \\ 0 & 0 & 0.4 & 0.4\end{pmatrix}\)
    2. (b) 0.3
    3. (c) 0.321

jmaths.xyz · AI HL · AHL 7.5 Unit review