Topic 4.19: Markov chains (HL only) Questions

To find the state vector 4 periods ahead you compute:

A region has 'urban' and 'rural' residents. Each year 5% of urban residents move rural and 10% of rural residents move urban; the rest stay. Write down the transition matrix T (order urban, rural).

T = [[0.7, 0.4], [0.3, 0.6]]. The probability of moving from state 2 to state 1 in one step is:

A column of a transition matrix is (0.6, 0.1, k)ᵀ. The value of k is:

Show that the long-term state is $\left(\begin{array}{c}400\\ 600\end{array}\right)$.

Which is NOT a valid transition matrix?

With T = [[0.5, 0.5], [0.5, 0.5]] and s₀ = (80, 20)ᵀ, the vector s₁ = T s₀ is:

Using T = [[0.95, 0.10], [0.05, 0.90]] and a start of 60 000 urban and 40 000 rural residents, find the numbers urban and rural after 1 year.

Write down the transition matrix, and the four probabilities it contains, with the town state first.

Find the state after one year and after two years.

Show that for any 2×2 transition matrix the column sums equal 1, and explain why a state vector's total stays constant after multiplying by T.

A subscription service has 'active' and 'lapsed' members with monthly T = [[0.85, 0.20], [0.15, 0.80]] (order active, lapsed). It starts with 10 000 active and 0 lapsed. Estimate the number active after 6 months and comment on the trend.

A library lends 'fiction' or 'non-fiction'. Each loan, 75% of fiction borrowers borrow fiction again and 40% of non-fiction borrowers switch to fiction; the rest stay. Find the transition matrix, then the proportions after 2 loans starting from 100% fiction.

An animal moves between feeding sites X, Y, Z with T = [[0.5, 0.2, 0.1], [0.3, 0.6, 0.4], [0.2, 0.2, 0.5]] (columns sum to 1). It starts at X with probability vector s₀ = (1, 0, 0)ᵀ. Find the probability it is at site Y after 2 moves.