Page 2 of 4 Midterm quiz for STAT3021 April 3, 2023 Write your answers clearly in ink, not in pencil. Except explicitly men- tioned, no justification is required. A. Suppose that{X n } n≥0 is a Markov Chain with state spaceS={1,2,3}and tran- sition matrix P=   1/21/20 a01−a 01−b b   where 0≤a≤1 and 0≤b≤1. Provide ALL possible numerical values for(a,b)in each of the situations in Questions 1 – 8. 2 marks each and 16 marks in total. 1. The chain is irreducible. 2. State 3 is absorbing. 3. State 3 has period 2. 4. State 3 is transient. Page 3 of 4 5. State 3 is recurrent. 6. The chain has a stationary distributionπ= (0,1/2,1/2). 7. The mean recurrence time of state 2 is∞. 8.f 33 = 1/2. Page 3 of 4 B. Let{X n ,n≥0}be a branching process with offspringξ 10 having the distribution: P(ξ 10 = 0) =aandP(ξ 10 = 3) = 1−a. The following Questions 9 -11 are related to this branching process. 3 marks each and 9 marks in total. 9. Find all values ofaso that the offspring varianceσ 2 is greater than the offspring meanμ. 10. Suppose thatX 0 = 1 anda >2/3. Find the extinction probabilityq. 11. Suppose thatX 0 = 1. Find all vlaues ofaso that the extinction probability q= 1/2. THIS IS THE LAST PAGE