IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on {1,2,3} has P(1,·)=(0.5,0.5,0), P(2,·)=(0.25,0.5,0.25), P(3,·)=(0,0.5,0.5). Starting in state 1, what is the expected number of steps to first reach state 3?
The expected time is 8 steps. Writing first-step equations m1 = 2 + m2 and m2 = 2 + 0.5m1 and solving gives m1 = 8 and m2 = 6.
- A4
- B5
- C6Correct
- D8
- 3
Explanation
Let m1, m2 be expected times to hit 3. m1 = 1 + 0.5m1 + 0.5m2 gives m1 = 2 + m2. m2 = 1 + 0.25m1 + 0.5m2 gives 0.5m2 = 1 + 0.25m1, so m2 = 2 + 0.5m1. Substituting: m1 = 2 + 2 + 0.5m1, so m1 = 8. Check: m2 = 6, and m1 = 2 + 6 = 8. So the answer is 8.
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