IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes
A time-homogeneous Markov chain on states {A, B} has one-step transition matrix with P(A→A)=0.7, P(A→B)=0.3, P(B→A)=0.4, P(B→B)=0.6. The chain starts in state A. What is the probability it is in state A after two steps?
The probability is 0.61. Sum over the intermediate state at step one: staying in A twice gives 0.49, and moving to B then back to A gives 0.12. Adding both paths gives 0.61, so omitting the path through B understates it.
- A0.49
- B0.52
- C0.55
- D0.58Correct
- 0.61
Explanation
Two-step probability P(A→A) = 0.7×0.7 + 0.3×0.4 = 0.49 + 0.12 = 0.61. Wait: check the sum over intermediate states A and B: via A 0.49, via B 0.12, total 0.61. Option 0.49 counts only the path through A.
Did you get it right without looking?
One question tells you little. A timed set on Stochastic processes shows your real accuracy, how long you take and where you lose marks.
More Stochastic processes questions
- Which of the following processes is a Markov process in continuous time with a discrete state space?
- Claims arrive at an Indian motor insurer as a Poisson process with rate 6 per hour. What is the probability that no claim arrives in a 20-mi…
- A two-state Markov jump process has states A (active) and B (inactive). The rate from A to B is 0.2 per year and from B to A is 0.6 per year…
- A simple random walk starts at 2 and moves +1 with probability 0.5 and -1 with probability 0.5 at each step. There are absorbing barriers at…
- A Markov chain on states {1,2,3,4} has transition probabilities p11 = 0.5, p12 = 0.5, p21 = 0.3, p22 = 0.7, p33 = 0.2, p34 = 0.8, p43 = 1. A…
- A simple random walk starts at 0. At each step it moves +1 with probability p and -1 with probability q = 1 - p, with steps independent. Whi…