课程名称︰机率导论 课程性质︰数学系大二必修 课程教师︰陈宏 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2015/06/18 考试时限（分钟）：60 试题 : (n) Notation: P = P(X = j| X = i) ij n 0 1. (20%) Three white and three black balls are distributed in two urns in such way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second urn, and conversely with the ball from the second urn. Let X denote n the state of the system after the n-th step. Explain why {X , n = 0, 1, 2, n ...} is a Markov chain and calculate its transition probability matrix. 2. Consider a Morkov chain with state space {A, B, C, D, E} and has transition matrix given by A B C D E A 0.1 0.6 0.3 0 0 B 0 0.2 0.3 0.5 0 C 0 0.3 0.2 0.3 0.2 D 0 0 0 0 1 E 0 0 0 1 0 (2) (n) (a) (10%) Calculate P and P for n ∈ |N. CA DE (b) (10%) What are the irreducible classes in this Markov chain? (c) (10%) Identify the recurrent and transient states. 3. (a) (10%) Let N = #{n ≧ 0, X = y} be the number of times that the process y n ∞ (n) is in state y. Show that E[N | X = y] = Σ P . y 0 n=0 yy ∞ (n) (b) (10%) Explain why a state y is recurrent if and only if Σ P = ∞. n=0 yy (c) (20%) (A Random Walk) Consider a Markov chain whose state space consists of the integers i = 0, ±1, ±2, ..., and has transition probabilities given by P = p, P = 1-p, i = 0, ±1, ±2, ... i, i+1 i, i-1 where 0 < p < 1. In other words, on each transition the process either moves one step to the right (with probability p) or one step to the left (with probability 1-p). Determine whether the state 0 is recurrent or transient. (Note that it depends on the values of p.) 4. An individual has three umbrellas, some at her office, and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will take an umbrella, if one is there. Otherwise, she gets wet. Assume that independent of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let X be the number of umbrellas at her n current location. (a) (15%) Find the transition probability for this Markov chain. (b) (15%) Calculate the limiting fraction of time she gets wet. -- 2 2 1 ψxavier13540 给定一个二次元(|R )上的开集 G，设 f: G →|R ∈ C 。考虑一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，则 ╰x(0) = x° t ω(x°) 只能是一定点、一周期轨道或连接有限个 critical point 的连通路径，不会像三 次元一样可能出现混沌(chaos)。此即为 ODE 动力系统中的 Poincaré–Bendixson 定理。 --

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