IEOR E4106: Stochastic Models (Spring 2024) Midterm Exam

发布时间：2024-05-09

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IEOR E4106: Stochastic Models (Spring 2024)

Midterm Exam

Problem 1 (50 points) Consider a Markov chain with initial value X0 = 3 and transition diagram shown in Figure 1. The numbers on arrows are transition probabilities.

1. (10 points) Is the Markov chain irreducible? Just give your answer without explanation.

2. (20 points) Compute Xn.

3. (20 points) Let T = min{n ≥ 1 : Xn = 3}. Compute E(T|X0 = 3).

Figure 1: Transition diagram.

Problem 2 (20 points) Let be a Markov chain with state space S = {1, 2} and transition probabilities P11 = 2/3 and P22 = 3/4. The initial state X0 is drawn uniformly at random from S. Let T = min{n ≥ 2 : Xn−2 = 1, Xn−1 = 2, Xn = 1}. Find ET.

Problem 3 (30 points) An alien visits one planet each year with Markovian transitions. If it comes to the Earth this year, then it will be here next year with probability A ∈ (0, 1), and there will be a UFO report this year with probability 0.9. If it does not come this year, then it will come next year with probability B ∈ (0, 1), and there will be a (false) UFO report this year with probability 0.3. Given its current location, the existence of UFO report this year is independent of the past. Historical data show that in the long run, 54% of the years have UFO reports.

1. (15 points) Find the proportion of years that the alien comes to the Earth.

2. (15 points) Let Zn = 1 if there are UFO reports in both the n-th and the (n − 1)-th year, and Zn = 0 otherwise. If Zn = 1 happens for 30.6% of the years, find A and B.

Your final answers to both questions should be numbers rather than expressions containing A or B.