IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A two-state chain on {1, 2} has P(1→2)=0.2 and P(2→1)=0.6. What is the stationary probability of state 1?
The stationary probability of state 1 is 0.75. Balancing flows gives 0.2π1 = 0.6π2, so π1 = 3π2, and normalising to sum to one gives π1 = 0.75 and π2 = 0.25.
- A0.25
- B0.40
- C0.60
- D0.75Correct
- 0.80
Explanation
Balance: π1(0.2)=π2(0.6), so π1=3π2. With π1+π2=1, π2=0.25 and π1=0.75. Check: 0.75×0.2=0.15=0.25×0.6. The option 0.25 comes from swapping the states.
Did you get it right without looking?
One question tells you little. A timed set on Markov chains shows your real accuracy, how long you take and where you lose marks.
More Markov chains questions
- A chain on {A,B,C} has P(A→B)=1, P(B→A)=0.5, P(B→C)=0.5, P(C→A)=1. What is the stationary probability of state A?
- A discrete-time stochastic process X_n on a finite state space satisfies P(X_{n+1}=j | X_n=i, X_{n-1}=i_{n-1}, ..., X_0=i_0) = P(X_{n+1}=j |…
- A No Claims Discount system for an Indian motor insurer has discount levels 0%, 20% and 40% (states 1, 2, 3). The annual probability of at l…
- A finite, irreducible, aperiodic Markov chain has stationary distribution π. Which statement is correct?
- In a Markov chain with transition matrix P, which condition must every row of P satisfy?
- A random walk starts at 0 and each step is +1 with probability p = 0.6 and -1 with probability 0.4, independently. What is the expected posi…