α 1−α P=1−ββ
Midterm quiz for STAT3021 April 19, 2021
1. (14 marks in total)
Suppose that {Xn}n≥0 is a Markov Chain with state space S = {0, 1} and transition
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where 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1. Provide All possible numerical values for (α,β) in each of the following Questions (a)-(g).
2 marks for each question.
(a) The chain is irreducible. (b) State 0 has period 2.
(c) State 0 is transient. (d) State 0 is recurrent.
(e) The chain has a stationary distribution π = (1/2, 1/2).
(f) The mean recurrence time μ0 of state 0 is 1.5 given that β = 0.
(g) f00 = 1/3
2. 11 marks in total
A gambler who has initial capital 20 dollars playing roulette makes a series of one dollar bets. He has a probability 0 < p < 1 of winning and 1 − p of losing each bet. The gambler decides to quit playing as soon as his net winning reach 25 dollars or his net losses reach 10 dollars.
(a) (4marks)DescribethecumulativefortuneofthegamblerbyaMCandprovide the state space of the MC.
(b) (2 marks) Find the probability that when he quits playing he will have won 25 dollars.
(c) (2 marks) In a fair play, i.e., p = 1/2, how many bets has the gambler placed on average when he quits playing?
(d) (3 marks) Suppose p < 1/2 and the gambler keeps playing without a quit rule. Find the probability that the gambler is ruined and the average number of the bets the gambler placed when he is ruined.
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