[응용통계] 15. 마코프 체인 (Markov Chains) (6) 연습문제

안녕하세요. 이번 포스팅에서는 마코프 체인에 대한 마지막 복습으로써, 연습문제를 가지고 왔습니다.

다들 완벽하게 마스터하시길 바랍니다.

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Quastion 1.

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A taxi driver provides service in two zones of a city. Fares picked up in zone A will have destinations in zone A with probability 0.6 or in zone B with probability 0.4. Fares picked up in zone B will have destinations in zone A with probability 0.3 or in zone B with probability 0.7. The driver’s expected profit for a trip entirely in zone A is 6; for a trip entirely in zone B is 8; and for a trip that involves both zones is 12. Find the taxi drivers` average profit per trip.

Quastion 2.

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Suppose that on each play of the game a gambler either wins 1 with probability 𝑝 or loses 1 with probability. 1 − 𝑝. The gambler continues betting until she or he is either up 𝑛 or down 𝑚. What is the probability that the gambler quits a winner?

Quastion 3.

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A particle moves among 𝑛 + 1 vertices that are situated on a circle in the following manner. At each step it moves one step either in the clockwise direction with probability 𝑝 or the counterclockwise direction with probability 𝑞 = 1 − 𝑝. Starting at a specified state, call it state 0, let 𝑇 be the time of the first return to state 0. Find the probability that all states have been visited by time 𝑇. Hint: Condition on the initial transition and then use results from the gambler’s ruin problem.

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