课程名称︰机率与统计 课程性质︰必修 课程教师︰谢宏昀 开课学院：电机资讯学院 开课系所︰电机工程学系 考试日期（年月日）︰104/4/30 考试时限（分钟）：180分钟 试题 : Probability and Statistics (Spring 2015) Midterm Exam PART I 1. 有一天，台大小鲁在 Dcard 看到一个故事： ────────────────────────────────────── 刚刚放学无聊准备去图书馆泡一下。我们学校图书馆前面有一排置物柜，我在那整理书 包。旁边 4 步左右的距离有一个男生也在整理，我恰巧看了一下旁边，发现我和他之 间的桌上放了一张学生证。 好人如我 我怕有人遗失找不到，就想拿去柜台放，就低头看了一下学生证。不看则已 ，一看不得了，是个清秀的帅哥啊啊啊。居然可以连学生证都拍那麽帅於是我不小心脱 口而出：「这麽帅乱丢学生证干嘛呢真是」。结果我听到笑声，抬头一看发现在我旁边 整理书包的男生就是学生证的主人?!?!?!?! 一整个神丢脸。 我整个尴尬说：「哎呀我不是故意看的啦＞＜」，直接冲进旁边的厕所躲起来。等到我 冷静下来不尴尬後走出厕所，发现学生证主人已经不在了（幸好呼）。我就准备继续整 理书包发现我书包旁边还是那张学生证 = 口 = 然後……我就放到柜台去了哈哈 ────────────────────────────────────── 这故事後来还有 Part 2, 3, 4, 5，後话不表，自己考後上网看。总之就是一整个超闪 到爆的故事。一直没有女友的小鲁，看完整个故事系列後，人生突然充满了希望，开始 跃跃欲试。 根据小鲁长年在图书馆的观察，在图书馆出没的女学生中，有七成是他喜欢的清秀佳人 型（是的，小鲁对女孩也是有他的品味跟坚持的，并不是只要是女生就行）。在图书馆 出没的女学生中，会有六成的人去置物柜区。小鲁也观察到，女孩子进洗手间再出来的 时间约是定值，都是五分钟左右（别问为什麽小鲁会观察这个，还不是因为你们考试的 需要！）。每天来置物柜区的人熙熙攘攘似是 Poisson，平均人流量 (rate) 大概是每 十分钟一人。这个学校的人眼睛很利，插在置物柜上的学生证一定都会看到。只要看到 有人遗失学生证，都会直接送去失物招领。另外小鲁的外表，嗯…相当有特色。通常女 生看到他的照片後，心中会觉得他「清秀」而脱口用「帅」形容他的，大约有一成。置 物柜每次使用投币十元一枚，使用完毕会退还。 小鲁看了狄卡闪文後，充满了希望，开始进行「脱鲁升温」大作战。他的剧本是：在置 物柜区徘徊，看到有女孩来，先判断是否是清秀佳人型的女孩。若然，他便先记好那女 孩放东西到哪格置物柜。接着便跟着女孩，远远观察这女孩念书，等到看到女孩念完书 要离开了，他再赶紧冲到置物柜把自己的学生证插在那女孩置物柜的隔壁格门缝。若女 孩看到了小鲁学生证没有漠视而且觉得小鲁照片清秀脱口说出他帅，小鲁再适时发出银 铃般的笑声（有点可怕…）。女孩会羞的进厕所，这时小鲁就把学生证插在女孩置物柜 门缝後走人，等着女孩从厕所後发现小鲁的学生证。若女孩从厕所出来前，没有其他多 事者把学生证送到失物招领，她便会拿到小鲁学生证，成为日後两人再次邂逅的契机！ (a) (7%) 小鲁在突然看到一个女生走进图书馆。试问在小鲁的「脱鲁升温」整套剧本 ，能在这女孩身上完全演练出来的机会是多少？（要进行到那个女生从厕所出来後 还能拿到小鲁学生证，才算完全演练出来） (b) (7%) 小鲁在图书馆待了一天，碰到了 30 个女生来。请问在 30 人中，出现 15 位女孩不是清秀佳人型、10 个清秀女孩有走到置物柜但漠视小鲁学生证的机率是 多少？ (c) (7%) 小鲁的室友，酸温，很瞧不起小鲁。呛声说「你这套剧本会有用，我温拿岂 不是白做的！」他甚至放话：「如果用这套剧本在10个来图书馆的女孩身上，真的 有女孩会因为看了你学生证觉得帅还听到你笑声会羞到躲到厕所去，有一个我就一 千块给你添行头，每多一个我就再加倍给（第二个 $2,000、第三个 $4,000、…） 。相反的，如果没有半个女孩演到躲厕所那步的话，你输给我 $1,000。敢不敢赌 ！」 请问，小鲁真的答应了酸温这个赌局，小鲁能从赌局赚到的钱是正是负？期 望值是多少？ (d) (7%) 小鲁在图书馆一上午按照剧本操作，对於来图书馆的 10 个女孩，剧本通通 都失败。请问在这条件之下，还要再额外多少个女孩来图书馆才会让小鲁剧本出现 第一次成功（要进行到那个女生从厕所出来後还能拿到小鲁学生证，才算成功）， 这额外的女孩个数的机率分布是什麽？ (e) (7%) 图书馆每天开馆时间是 14.5 小时。自从打定主意後，小鲁每天都是图书馆 一开门就冲到图书馆置物柜区的人。小鲁相当迷信，一直认为「3」是他的幸运数 字，所以他对於每天开馆後第三个到置物柜区的人特别有好感（第三个是不把小鲁 算在内），认为是他的幸运者。请问今天当小鲁一开馆就冲到了置物柜区，从他今 天花在等待幸运者所花的时间，其 CDF 是什麽呢？ ────────────────────────────────────── 经过一周的努力，小鲁终於遇上了梦中的女孩，小鲁告白成功。在这重要时刻，阴险的 酸温没办法忍受寝室内还有其他温拿，於是决定要恶意的跟女孩透露小鲁与他的赌局。 酸温的诡计能否得逞？女孩是否会反悔？小鲁能否继续守护着这得来不易的爱情？ 这些，都已经都跟机率无关了。大家，就安心的考试吧～ PART II 2. (12%) Peter throws 3 dices, each with 4 faces. The probability of each face (i.e., 1, 2, 3 and 4) is 1/4. (a) (6%) Please calculate the PMF of the sum of the 3 dices. (Hint: there are only 64 possible outcomes) (b) (6%) Denote the sum of the 3 dices as X, X is a random variable. Please calculatethe median, mode and expected value of X. (2% for each answer) 3. (8%) Let X be an exponential random variable with a mean of 1/λ. (a) (4%) Please calculate the conditional PDF of X given X > T, where T is a positive real number. (b) (4%) Let Y = X - T, given X > T. What are the PDF and expected value of Y, given X > T? (2% for each answer) (c) (0%) What can you conclude from your answers? 4. (15%) Let X be the number of earthquakes in T units of time, starting from time 0. Assume that X follows a Poisson distribution with a mean of λT. It is known that the NTU MD building is equipped with an earthquake detector that can detect an earthquake with a probability p. That is, for each earthquake, the detector may miss-detect the earthquake with a probability of 1 - p. (a) (10%) Let Y be the nurnber of detected earthquakes in T units of time. What is the PMF of Y? (Hint: You may start from the conditional PMF first) (b) (5%) Let D be the time interval, starting from time 0, that the first miss-detected earthquake occurs. If an earth quake is miss detected, we all die. What is thePDF of our life span, counting down from time 0? (Hint: Your answer of Y should help here. Calculate the CDF of the life span first) PART III 5. (12%) Sometimes it is difficult to find the probability of the desired event directly. In this situation, finding probability bounds could be helpful. (a) (4%) Let {A_i|i = 1,2,...,n} be a collection of events. Prove the following inequality: http://ppt.cc/uh82 (b) (4%) Given a set of n > 1 nodes on a 2-D plane, generate random graphs by connecting each pair of nodes with probability p, 0 < p < 1, independently of others in the graph. Let B_n be the event that a random graph of n nodes hasat least one isolated node (i.e., a node that is not connected to any other nodes). Use (a) to show that n-1 P[B_n] ≦ n(1 - p) . (c) (4%) Let p = (1 + ε)(ln n)/n, where ε > 0. Find the Probability of having a connected random gragh (i.e. no isolated nodes) as the number of nodes n → ∞. 6. (10%) Consider the RLC circuit shown in the following figure. http://ppt.cc/oa6a It is known that the voltage transfer function between the source (sinusoidal source with radian frequency ω) and the capacitor C can be written as http://ppt.cc/R4cw If R < √(2L/C), the RLC circuit has resonance and the resonant frequency of the circuit ω0, where |H(ω)|^2 attains its maximum, is http://ppt.cc/BT8Z In a microelectronic circuit lab, it is required that the components with R = 1 (ohm), L = 1 (henry), and C = 1 (farad) are used. Due to fabrication issue, however, resistors is 1 and the variance is 1/3. It is reasonable to assume that the value of R follows a (continuous) uniform distribution for lack of a more accurate probability model. (a) (2%) You randomly pick a resistor from the 1-ohm bin to connect the RLC circuit. Find the probability that your RLC circuit can have resonance. (b) (3%) For those RLC circuits with resonance, find the probability that the resonant frequency ω0 is between 1/√2 and 1. (c) (5%) For those RLC circuits with resonance, find the PDF of the resonant peak |H(ω0)|^2. 7. (13%) Generating samples of random variables is one important function during computer simulation. However, by default the computer provides only (continuous) uniform (0, 1) random variable (called it U), and hence it is important to transform these samples to fit the distribution of the desired random variable. (a) (5%) Although the CDF of any randorn variable is a non-decreasing function, sometimes it may contain jumps (discontinuities) and "flat" intervals as shown inthe following figure. http://ppt.cc/8VD6 For any general CDF F_X(x) as such, define a new function as follows: ~ F(u) = min{x|F_X(x) ≧ u}, for all 0 < u < 1. ~ Show derived random variable Y = F(U) has CDF F_Y(x) = F_X(x). (b) (4%) Describe the procedure of generating a geometric (p) random variable (0 < p < 1) from U. You should clearly write out all mathematical equations used in the procedure. Note that; you can use (a) or any other methods to do the transformation. (c) (4%) Sometimes it may be difficult to directly apply (a) to do the transformation. Instead, knowledge about the mathematical property of a random variable may be helpful in generating the samples. Based on what you have learned about the Poisson random variable, describe the procedure of generating a Poisson (α) random variable (a > 0) from U. (Hint: Find a random variable that can be easily transformed from U and then generate samples of the Poisson distribution based on this intermediate random variable.) --

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