Stationary Distributions and Long-Run Behaviour · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Where a random walk settles down

Mathematics · Probability · ages 22-24
Name ______________________   Date ____________

  1. An absorbing state loops to itself with probability 1. Is its period 1?

    Circle one:   True   False

  2. A two-state chain has transition matrix with rows [0.8, 0.2] and [0.3, 0.7]. What is the stationary probability of state 0?

    Answer: ______________

  3. A chain flips between two states every step: from 0 it always goes to 1 and back. How is it classified?

    • Irreducible and aperiodic
    • Reducible
    • Irreducible but periodic
    • Absorbing

  4. A student finds a stationary vector (0.6, 0.4) and checks that it sums to one. That check is the right sanity test.

    Circle one:   True   False

  5. A periodic chain need not converge, yet it can still have a stationary distribution.

    Circle one:   True   False

  6. A two-state chain has rows [0.7, 0.3] and [0.2, 0.8]. What is the stationary probability of state 1?

    Answer: ______________

  7. A periodic chain need not converge, yet Mia claims it can still have a stationary distribution. Is Mia right?

    Circle one:   True   False

  8. A chain on {0, 1, 2} moves between 0 and 1 freely but state 2 loops to itself always. Is it irreducible?

    • No, because state 2 can never be left
    • Yes, every state is reachable
    • No, because all states are absorbing
    • Yes, by symmetry

  9. A two-state chain has rows (0.7, 0.3) and (0.2, 0.8). What is the stationary probability of state 1?

    Answer: ______________

  10. A chain has rows (0.5, 0.5) and (0.25, 0.75) and starts in state 0. What is the chance of being in state 1 after two steps?

    • 0.5
    • 0.625
    • 0.75
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Where a random walk settles down W1-mt_MGfHvUGkeh-s1

  1. True · Once entered it is never left, so returns happen at every step and the period is 1.
  2. 0.6 · From pi = pi P, 0.2 pi_0 = 0.3 pi_1 with pi_0 + pi_1 = 1 gives pi_0 = 0.6.
  3. Irreducible but periodic · Each state reaches the other, but returns take an even number of steps, so the chain is irreducible but periodic.
  4. True · A stationary distribution is a probability vector, so its entries must sum to one.
  5. True · The flip-flop chain has stationary vector (0.5, 0.5) though its powers oscillate.
  6. 0.6 · Here pi_1 = P_01 / (P_01 + P_10) = 0.3/0.5 = 0.6.
  7. True · The flip-flop chain has stationary vector (0.5, 0.5) even though its powers oscillate forever.
  8. No, because state 2 can never be left · State 2 traps the chain, so states 0 and 1 cannot be reached from it: no, because state 2 can never be left.
  9. 0.6 · Balance of flows gives 0.3 over 0.5 for state 1.
  10. 0.625 · After one step the distribution is (0.5, 0.5); one more step gives 0.25 plus 0.375.
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