※ [本文转录自 NTU-Exam 看板] 作者: ketsu1109 (德州安打) 看板: NTU-Exam 标题: [试题] 98上 叶丙成 随机程序及应用 2nd期中考 时间: Wed Dec 16 20:07:45 2009 课程名称︰随机程序及应用 课程性质︰选修 课程教师︰叶丙成 开课学院：电资学院 开课系所︰电信所 考试日期（年月日）︰2009.12.16 考试时限（分钟）：180+40(延长) 是否需发放奖励金：是，谢谢 （如未明确表示，则不予发放） 试题 : 1.(1%) Please write down your name on the top of every answer sheet. 2.(以下情节纯属虚构) 在某校後门附近有一家餐厅「没钱真惨」，餐厅内共有四桌，生意相当不错。一旦满 　桌，新来的客人通常直接去相邻的「没钱卡是惨」餐厅用餐。故「没钱卡是惨」老板 　对於「没钱真惨」的生意状况相当在意，时常派老板娘偷偷打探。 　根据「没钱卡是惨」老板娘的观察，在中午以十五分钟为单位时间，「没钱真惨」在 　每单位时间内有一组客人去吃饭的机率为p1，有两组客人上门之机率为p2。老板娘鲜 　少见到十五分钟内来超过两组客人的情形发生（Hint: 由此可知没有人来的机率？）。 　同组客人多为某校同研究室学生，这些学生习性多是坐於同桌用餐。用餐时间平均需 　要一个单位时间左右，因此鲜少在到达的该单位时间内用完餐。 　「没钱卡是惨」老板娘发现到某校学生用餐习性实在相当糟糕，用完餐後常常坐着聊 　天，对於自己已占桌多久似乎完全没有记忆。往往都是同桌学生有人想到该离开了才 　会整桌离开。平均而言，在到达餐厅後的接下来的每一个单位时间内同桌学生中会有 　人想到该离开餐厅的机率为 q（Hint: 若有 n桌客人，每单位时间有 k桌人离开之机 率为何种分布？）。 (a) (10%) 让「没钱卡是惨」的老板相当在意的是「没钱真惨」每个单位时间中客人 的总桌数，某日某生与同学去「没钱卡是惨」用餐聊天时被老板听到正在修随机 　　　程序。老板心中大喜，拜托某生帮他建立「没钱真惨」用餐桌数的马可夫模型。 　　　请你帮某生画出该模型的 state diagram。 (b) ( 5%) 「没钱真惨」老板对於学生占桌聊天，致使满桌後流失客人到「没钱卡是 　　　惨」，内心耿耿於怀，对此心生一计。每当「没钱真惨」满桌时，餐厅内冷气常 　　　会莫名的故障，直到客人桌数在两桌以下（包含两桌）才恢复正常。某校学生相 　　　当不耐热，冷气故障後会想到要离开的机率成为两倍2q。请你帮某生画出新模型 　　　的 state diagram。 　(c) ( 5%) 「没钱卡是惨」老板决定跟「没钱真惨」杠上了，只要老板娘发现「没钱 　　　真惨」用餐桌数有三桌以上（包含三桌），「没钱卡是惨」便会推出五折大优惠 　　　的活动，直到「没钱真惨」降到两桌以下（包含两桌）才恢复正常。某校学生相 　　　当贪小便宜，对餐厅毫无忠诚度可言。在「没钱卡是惨」有五折大优惠时会去「 　　　没钱真惨」用餐的机率大减成为每单位时间内有一组客人上门之机率为 0.1p1， 有两组客人上门之机率为 0.1p2。请你帮某生画出新模型的 state diagram。（ 　　　Note: 冷气搞鬼依然不变） 　(d) ( 5%) （加分题，建议有时间再做） 　　　「没钱卡是惨」老板娘每次看到「没钱真惨」满桌，心情就会很恶劣。回来店内 　　　往往对「没钱卡是惨」老板呼来喝去，手打脚踢，让老板苦不堪言。老板娘的恶 　　　劣心情往往要等到「没钱真惨」没有满桌後才会恢复正常。「没钱卡是惨」老板 　　　为有效掌握老板娘在各单位时间内之心理状态，另委由某生帮他建立老板娘心理 　　　状态之 two-state（正常、恶劣）马可夫模型。请你帮某生画出新模型的 state diagram。 （Note: 冷气搞鬼依然不变） 3. A machine consists of two components A and B, and the machine will function only if both of the components are working. Assume that the lifetime of the two components are independent exponential random variables with rate 1 for A and 3 for B. The machine is working at time t=0. Once a component fails, a new component is replaced immediately. (a) (4%) Find the expected time until the first failure. (b) (4%) Find the variance of the time until the first failure. (c) (4%) Find the probability that there are no failures before time t=T. (d) (4%) Given that there are no failures until time t=T, determine the conditional probability that the first replacement is for component A. (e) (4%) Find the probability that the machine fails before t=T and it is A that is the cause of the failure. 4. Let { N(t),t≧0 } be a Poisson process of rate λ. (a) (4%) Find the probability of the event { N(t)=n }. (b) (4%) Find Cov( N(t),N(t+s) ), s≧0. (c) (4%) Find the probability of the event { N(1)=1, N(2)=2 }. (d) (4%) Find P{ N(s)=n1 | N(t)=n2 }, t≧s. (e) (4%) Given that only one event happens before t=T, find the conditional distribution of the time when that event happens. 5. Buses arrive at a certain stop according to a Poisson process with rate λ/minute. If you take the bus from that stop then it takes R minutes, measured from the time at which you enter the bus, to arrive home. If you walk from the bus stop then it takes W minutes to arrive home. Suppose that your policy when arriving at the bus stop is to wait up to S minutes, and if a bus has not yet arrived by that time then you walk home. (a) (4%) Suppose that you always take a bus home; that is, you choose S=∞. What is the expected time from when you arriving at the bus stop until you reach home? (b) (4%) Compute the expected time from when you arrive at the bus stop until you reach home. Your answer should be a function of S. (c) (4%) Find the optimal value of S that minimizes the expected time of part (b). 6. A taxi travels between three locations. When it reaches location 1 it will go next to 2 or 3 with probability 2/3 and 1/3, respectively. When it reaches location 2 it will go next to 1 with probability 1/3 and to 3 with probability 2/3. From 3, it will go to 1 and 2 with probability 2/3 and 1/3, respectively. The mean traveling times between locations i and j are t12=20, t13=30, t23=30 (tij=tji). Upon arrival at a location the taxi immediately departs. (a) (4%) What is the limiting probability that the taxi's most recent stop was at location i, i=1,2,3? (b) (4%) What is the limiting probability that the taxi is heading for location 2? (c) (4%) What fraction of time is the taxi traveling from 2 to 3 in the long run? 7. The weather tomorrow in Taipei on a given day is correlated to the previous weather conditions of today and yesterday. If the weather is sunny today and yesterday, it will be sunny tomorrow with probability 0.7. If the weather is rainy today and sunny yesterday, it will be sunny tomorrow with probability 0.4. If the weather is sunny today and rainy yesterday, it will be sunny tomorrow with probability 0.6. If the weather is rainy today and yesterday, it will be sunny tomorrow with probability 0.3. Consider the situation that it is sunny today and yesterday. (a) (4%) Plot the minimum state Markov chain model for this weather tran- sition. Please be sure that all the transion probabilities are marked and each of the states is clearly defined. (b) (4%) Evaluate the steady state probabilities. (c) (4%) Let X be the number of days up to, but not including the first future rainy day. Find the PMF of X. (d) (4%) Find the probability of seeing the forst future rainy day followed by another rainy day. -- ┌这篇文章让你觉得?∮weissxz ──────────────────────┐ │ █●█ █●█ █●█ █●█ █●█ █●█ █●█ │ │ ‵ ′ ‵ ′ ‵ ′ "‵ ′$ ‵ ′ ‧ ‧ ◎ ◎ │ │ " ﹏ " " ︺ " ////// △ / " ︺ " 皿 │ │ 新奇 。温馨。 ＊害羞＊ $俭朴$ #靠夭# +闪酿+ 炸你家 │ └────────────────────────────────────┘ --

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