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

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

  1. A state A has transition probability P(A, A) = 1. Its period is 1.

    Circle one:   True   False

  2. A chain uses a fixed transition matrix and has stationary distribution (0.35, 0.65), in state order (0, 1). It starts in that distribution. What is the probability of state 0 after 50 steps?

    Answer: ______________

  3. Use state order (0, 1, 2). A chain has transition rows (0.5, 0.5, 0), (0, 0, 1), (1, 0, 0). How is it classified?

    • Reducible, with period 1
    • Irreducible, with period 1
    • Irreducible, with period 3
    • Reducible, with period 3

  4. Checking that a candidate stationary distribution (0.6, 0.4) sums to 1 is necessary, but that check alone does not prove stationarity.

    Circle one:   True   False

  5. A finite periodic chain can have a stationary distribution even though its transition matrix powers do not converge.

    Circle one:   True   False

  6. Use state order (0, 1). A chain has stationary distribution (0.25, 0.75) and transition rows (1 - r, r), (0.1, 0.9). What is r?

    Answer: ______________

  7. Use state order (0, 1). The fixed transition rows (1, 0), (0.4, 0.6) define a reducible chain, but every starting distribution approaches (1, 0).

    Circle one:   True   False

  8. Use state order (0, 1). A chain has the identity transition matrix, with rows (1, 0), (0, 1). Which describes all its stationary distributions?

    • Every (a, 1 - a) with 0 <= a <= 1
    • Only (0.5, 0.5), because both states must have equal weight
    • Only (1, 0), because state 0 is listed before state 1
    • No vector, because the two states cannot reach one another

  9. Use state order (0, 1, 2) for rows and columns. A chain has fixed transition rows (1, 0, 0), (0.15, 0.4, 0.45), (0, 0, 1). Starting at state 1, what does the probability of state 2 approach as the number of steps grows? Give a decimal.

    Answer: ______________

  10. A chain uses fixed transition rows (0.5, 0.5), (0.25, 0.75), with row and column order (0, 1). Starting at state 0, what is the probability of state 1 after two steps?

    • 0.500
    • 0.625
    • 0.750
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Answer key

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Stationary weights and limiting distributions W1-mt_MGfHvUGkeh-s1

  1. True · A is absorbing. Every positive integer is a possible return time, including 1, so their greatest common divisor is 1.
  2. 0.35 · A stationary starting distribution stays unchanged at every step, so the state 0 probability remains 0.35.
  3. Irreducible, with period 1 · The positive-probability cycle connects every state. State 0 has a self-loop, giving period 1 throughout this irreducible chain.
  4. True · Normalization is necessary. Also check nonnegative entries and pi P = pi.
  5. True · The deterministic flip preserves (0.5, 0.5), but its matrix powers alternate between the flip matrix and the identity.
  6. 0.3 · Stationary flow balance gives 0.25r = 0.75 × 0.1 = 0.075, so r = 0.3.
  7. True · State 0 is absorbing. The state 1 weight after n steps is its starting weight times 0.6^n, which tends to 0.
  8. Every (a, 1 - a) with 0 <= a <= 1 · The identity matrix leaves every row vector unchanged. Nonnegative entries totaling 1 give exactly the stated family.
  9. 0.75 · States 0 and 2 are absorbing. From state 1, the limiting state 2 probability satisfies h = 0.45 + 0.4h, so h = 0.75.
  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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