IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on states {1,2} has one-step transition matrix with rows (0.7, 0.3) for state 1 and (0.4, 0.6) for state 2. The chain starts in state 1 at time 0. What is P(X_2 = 2)?
The probability is 0.39. Summing over the intermediate state at time 1 gives 0.7×0.3 for staying then moving, plus 0.3×0.6 for moving then staying, which equals 0.21 plus 0.18, the (1,2) entry of the two-step matrix.
- A0.30
- B0.39
- C0.45Correct
- D0.51
- 0.60
Explanation
P(X_2=2) = P(1→1)P(1→2) + P(1→2)P(2→2) = 0.7×0.3 + 0.3×0.6 = 0.21 + 0.18 = 0.39. Check via the squared matrix: entry (1,2) = 0.7×0.3 + 0.3×0.6 = 0.39. Option 0.45 is wrong because it does not follow from this sum.
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