[試題] 98上 葉丙成 隨機程序及應用 2nd期中考

[ketsu1109 (德州安打)]

課程名稱︰隨機程序及應用
課程性質︰選修
課程教師︰葉丙成
開課學院：電資學院
開課系所︰電信所
考試日期（年月日）︰2009.12.16
考試時限（分鐘）：180+40(延長)
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）

試題 :

1.(1%) Please write down your name on the top of every answer sheet.

2.(以下情節純屬虛構)
  在某校後門附近有一家餐廳「沒錢真慘」，餐廳內共有四桌，生意相當不錯。一旦滿
　桌，新來的客人通常直接去相鄰的「沒錢卡是慘」餐廳用餐。故「沒錢卡是慘」老闆
　對於「沒錢真慘」的生意狀況相當在意，時常派老闆娘偷偷打探。

　根據「沒錢卡是慘」老闆娘的觀察，在中午以十五分鐘為單位時間，「沒錢真慘」在
　每單位時間內有一組客人去吃飯的機率為p1，有兩組客人上門之機率為p2。老闆娘鮮
　少見到十五分鐘內來超過兩組客人的情形發生（Hint: 由此可知沒有人來的機率？）。
　同組客人多為某校同研究室學生，這些學生習性多是坐於同桌用餐。用餐時間平均需
　要一個單位時間左右，因此鮮少在到達的該單位時間內用完餐。

　「沒錢卡是慘」老闆娘發現到某校學生用餐習性實在相當糟糕，用完餐後常常坐著聊
　天，對於自己已佔桌多久似乎完全沒有記憶。往往都是同桌學生有人想到該離開了才
　會整桌離開。平均而言，在到達餐廳後的接下來的每一個單位時間內同桌學生中會有
　人想到該離開餐廳的機率為 q（Hint: 若有 n桌客人，每單位時間有 k桌人離開之機
  率為何種分佈？）。

  (a) (10%) 讓「沒錢卡是慘」的老闆相當在意的是「沒錢真慘」每個單位時間中客人
      的總桌數，某日某生與同學去「沒錢卡是慘」用餐聊天時被老闆聽到正在修隨機
　　　程序。老闆心中大喜，拜託某生幫他建立「沒錢真慘」用餐桌數的馬可夫模型。
　　　請你幫某生畫出該模型的 state diagram。

  (b) ( 5%) 「沒錢真慘」老闆對於學生佔桌聊天，致使滿桌後流失客人到「沒錢卡是
　　　慘」，內心耿耿於懷，對此心生一計。每當「沒錢真慘」滿桌時，餐廳內冷氣常
　　　會莫名的故障，直到客人桌數在兩桌以下（包含兩桌）才恢復正常。某校學生相
　　　當不耐熱，冷氣故障後會想到要離開的機率成為兩倍2q。請你幫某生畫出新模型
　　　的 state diagram。

　(c) ( 5%) 「沒錢卡是慘」老闆決定跟「沒錢真慘」槓上了，只要老闆娘發現「沒錢
　　　真慘」用餐桌數有三桌以上（包含三桌），「沒錢卡是慘」便會推出五折大優惠
　　　的活動，直到「沒錢真慘」降到兩桌以下（包含兩桌）才恢復正常。某校學生相
　　　當貪小便宜，對餐廳毫無忠誠度可言。在「沒錢卡是慘」有五折大優惠時會去「
　　　沒錢真慘」用餐的機率大減成為每單位時間內有一組客人上門之機率為 0.1p1，
      有兩組客人上門之機率為 0.1p2。請你幫某生畫出新模型的 state diagram。（
　　　Note: 冷氣搞鬼依然不變）

　(d) ( 5%) （加分題，建議有時間再做）
　　　「沒錢卡是慘」老闆娘每次看到「沒錢真慘」滿桌，心情就會很惡劣。回來店內
　　　往往對「沒錢卡是慘」老闆呼來喝去，手打腳踢，讓老闆苦不堪言。老闆娘的惡
　　　劣心情往往要等到「沒錢真慘」沒有滿桌後才會恢復正常。「沒錢卡是慘」老闆
　　　為有效掌握老闆娘在各單位時間內之心理狀態，另委由某生幫他建立老闆娘心理
　　　狀態之 two-state（正常、惡劣）馬可夫模型。請你幫某生畫出新模型的 state
      diagram。 （Note: 冷氣搞鬼依然不變）


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.

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