IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A Markov chain on states {1,2,3,4} has transition matrix rows: P(1,·)=(0.5,0.5,0,0); P(2,·)=(0.4,0.6,0,0); P(3,·)=(0,0,0.3,0.7); P(4,·)=(0,0,0.6,0.4). Which statement about its communicating classes is correct?
There are two closed communicating classes, {1,2} and {3,4}. Each pair communicates internally and no probability leaks out of either block, so both are recurrent and the chain is reducible.
- AThe chain is irreducible because every state has a positive self-transition probability
- BThere are two closed communicating classes, {1,2} and {3,4}Correct
- CThere are two communicating classes, and {1,2} is transient
- DThere is one closed class {1,2} and {3,4} is transient
- There are four communicating classes, one for each state
Explanation
States 1 and 2 reach each other and never leave {1,2}. States 3 and 4 likewise form a closed set. So there are two closed classes, both recurrent. Self-transitions do not link different blocks, so the chain is not irreducible.
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