IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov processes
A Markov chain on states 1, 2, 3 has transition matrix rows: 1: (0.5, 0.5, 0); 2: (0.25, 0.5, 0.25); 3: (0, 0.5, 0.5). What is the stationary probability of state 2?
The stationary probability of state 2 is 0.5. Balance at state 1 gives π1 = 0.5π2, and symmetry gives π3 = 0.5π2, so the probabilities 0.25, 0.5, 0.25 sum to one.
- A0.25
- B1/3
- C0.5Correct
- D0.6
- 0.4
Explanation
Equation for state 1: π1 = 0.5π1 + 0.25π2, so π1 = 0.5π2. By symmetry π3 = 0.5π2. Then 0.5π2 + π2 + 0.5π2 = 1 gives π2 = 0.5. Check state 2: 0.5·0.5 + 0.5·0.5 + 0.5·0.25... equals 0.25+0.25+... consistent with π2 = 0.5 via π1 = π3 = 0.25 .
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