課程名稱︰機率與統計
課程性質︰必修
課程教師︰謝宏昀
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰104/4/30
考試時限（分鐘）：180分鐘
試題 :
Probability and Statistics (Spring 2015) Midterm Exam
PART I
1. 有一天，台大小魯在 Dcard 看到一個故事：
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剛剛放學無聊準備去圖書館泡一下。我們學校圖書館前面有一排置物櫃，我在那整理書
包。旁邊 4 步左右的距離有一個男生也在整理，我恰巧看了一下旁邊，發現我和他之
間的桌上放了一張學生證。
好人如我 我怕有人遺失找不到，就想拿去櫃檯放，就低頭看了一下學生證。不看則已
，一看不得了，是個清秀的帥哥啊啊啊。居然可以連學生證都拍那麼帥於是我不小心脫
口而出：「這麼帥亂丟學生證幹嘛呢真是」。結果我聽到笑聲，抬頭一看發現在我旁邊
整理書包的男生就是學生證的主人?!?!?!?! 一整個神丟臉。
我整個尷尬說：「哎呀我不是故意看的啦＞＜」，直接衝進旁邊的廁所躲起來。等到我
冷靜下來不尷尬後走出廁所，發現學生證主人已經不在了（幸好呼）。我就準備繼續整
理書包發現我書包旁邊還是那張學生證 = 口 = 然後……我就放到櫃檯去了哈哈
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這故事後來還有 Part 2, 3, 4, 5，後話不表，自己考後上網看。總之就是一整個超閃
到爆的故事。一直沒有女友的小魯，看完整個故事系列後，人生突然充滿了希望，開始
躍躍欲試。
根據小魯長年在圖書館的觀察，在圖書館出沒的女學生中，有七成是他喜歡的清秀佳人
型（是的，小魯對女孩也是有他的品味跟堅持的，並不是只要是女生就行）。在圖書館
出沒的女學生中，會有六成的人去置物櫃區。小魯也觀察到，女孩子進洗手間再出來的
時間約是定值，都是五分鐘左右（別問為什麼小魯會觀察這個，還不是因為你們考試的
需要！）。每天來置物櫃區的人熙熙攘攘似是 Poisson，平均人流量 (rate) 大概是每
十分鐘一人。這個學校的人眼睛很利，插在置物櫃上的學生證一定都會看到。只要看到
有人遺失學生證，都會直接送去失物招領。另外小魯的外表，嗯…相當有特色。通常女
生看到他的照片後，心中會覺得他「清秀」而脫口用「帥」形容他的，大約有一成。置
物櫃每次使用投幣十元一枚，使用完畢會退還。
小魯看了狄卡閃文後，充滿了希望，開始進行「脫魯升溫」大作戰。他的劇本是：在置
物櫃區徘徊，看到有女孩來，先判斷是否是清秀佳人型的女孩。若然，他便先記好那女
孩放東西到哪格置物櫃。接著便跟著女孩，遠遠觀察這女孩唸書，等到看到女孩唸完書
要離開了，他再趕緊衝到置物櫃把自己的學生證插在那女孩置物櫃的隔壁格門縫。若女
孩看到了小魯學生證沒有漠視而且覺得小魯照片清秀脫口說出他帥，小魯再適時發出銀
鈴般的笑聲（有點可怕…）。女孩會羞的進廁所，這時小魯就把學生證插在女孩置物櫃
門縫後走人，等著女孩從廁所後發現小魯的學生證。若女孩從廁所出來前，沒有其他多
事者把學生證送到失物招領，她便會拿到小魯學生證，成為日後兩人再次邂逅的契機！
(a) (7%) 小魯在突然看到一個女生走進圖書館。試問在小魯的「脫魯升溫」整套劇本
，能在這女孩身上完全演練出來的機會是多少？（要進行到那個女生從廁所出來後
還能拿到小魯學生證，才算完全演練出來）
(b) (7%) 小魯在圖書館待了一天，碰到了 30 個女生來。請問在 30 人中，出現 15
位女孩不是清秀佳人型、10 個清秀女孩有走到置物櫃但漠視小魯學生證的機率是
多少？
(c) (7%) 小魯的室友，酸溫，很瞧不起小魯。嗆聲說「你這套劇本會有用，我溫拿豈
不是白做的！」他甚至放話：「如果用這套劇本在10個來圖書館的女孩身上，真的
有女孩會因為看了你學生證覺得帥還聽到你笑聲會羞到躲到廁所去，有一個我就一
千塊給你添行頭，每多一個我就再加倍給（第二個 $2,000、第三個 $4,000、…）
。相反的，如果沒有半個女孩演到躲廁所那步的話，你輸給我 $1,000。敢不敢賭
！」 請問，小魯真的答應了酸溫這個賭局，小魯能從賭局賺到的錢是正是負？期
望值是多少？
(d) (7%) 小魯在圖書館一上午按照劇本操作，對於來圖書館的 10 個女孩，劇本通通
都失敗。請問在這條件之下，還要再額外多少個女孩來圖書館才會讓小魯劇本出現
第一次成功（要進行到那個女生從廁所出來後還能拿到小魯學生證，才算成功），
這額外的女孩個數的機率分布是什麼？
(e) (7%) 圖書館每天開館時間是 14.5 小時。自從打定主意後，小魯每天都是圖書館
一開門就衝到圖書館置物櫃區的人。小魯相當迷信，一直認為「3」是他的幸運數
字，所以他對於每天開館後第三個到置物櫃區的人特別有好感（第三個是不把小魯
算在內），認為是他的幸運者。請問今天當小魯一開館就衝到了置物櫃區，從他今
天花在等待幸運者所花的時間，其 CDF 是什麼呢？
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經過一週的努力，小魯終於遇上了夢中的女孩，小魯告白成功。在這重要時刻，陰險的
酸溫沒辦法忍受寢室內還有其他溫拿，於是決定要惡意的跟女孩透露小魯與他的賭局。
酸溫的詭計能否得逞？女孩是否會反悔？小魯能否繼續守護著這得來不易的愛情？
這些，都已經都跟機率無關了。大家，就安心的考試吧～
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.)