[응용통계] 10. 마코프 체인 (Markov Chains) (4) 연습문제

안녕하세요, 이번 포스팅에서는 지금까지 배운 부분에 대하여 연습문제를 풀어보도록 하겠습니다.

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Quastion 1.

Suppose that whether or not it rains today depends on previous weather conditions through the last three days. Show how this system may be analyzed by using a Markov chain. How many states are needed?

풀이)

R : 비가 올 경우의 수, U : 비가 오지 않을 경우의 수하고 정의하자.

다음 날의 비가 오거나 그렇지 않을 확률은 이전 3일동안의 결과에 따라 달라지므로, 3일동안 가능한 경우의 수는 RRR, RRU, RUR, RUU, URR, URU, UUR, UUU 총 8가지가 된다. 따라서 State는 총 8개가 필요하고, 미래의 시점의 비가 올 확률은 그 이전 state로부터만 영향을 받으므로 Markov chain이라고 할 수 있다. 위 문제의 Markov chain을 그림으로 표현하면 다음과 같다.

Quastion 2.

In Problem 1, suppose that if it has rained for the past three days, then it will rain today with probability 0.8; if it did not rain for any of the past three days, then it will rain today with probability 0.2; and in any other case the weather today will, with probability 0.6, be the same as the weather yesterday. Determine P for this Markov chain.

풀이)

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Quastion 3.

Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has probability 0.6 of coming up heads. If the coin flipped today comes up heads, then we select coin 1 to flip tomorrow, and if it comes up tails, then we select coin 2 to flip tomorrow. If the coin initially flipped is equally likely to be coin 1 or coin 2, then what is the probability that the coin flipped on the third day after the initial flip is coin 1? Suppose that the coin flipped on Monday comes up heads. What is the probability that the coin flipped on Friday of the same week also comes up heads?

풀이)

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Coin에 대한 Markov chain을 그림으로 나타내면 다음과 같다.

Quastion 4~5.

A store stocks a particular item. The demand for the product each day is 1 item with probability 1/6, 2 items with probability 3/6, and 3 items with probability 2/6. Assume that the daily demands are independent and identically distributed. Each evening if the remaining stock is less than 3 items, the store orders enough to bring the total stock up to 6 items. These items reach the store before the beginning of the following day. Assume that any demand is lost when the item is out of stock.

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