课程名称︰机率
课程性质︰必修
课程教师︰林守德
开课学院：电机资讯
开课系所︰资讯工程
考试日期（年月日）︰2014/6/16
考试时限（分钟）：180
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
‧ Note, please use Φ function as the CDF of standard normal distribution (no
need to calculate the correct value). For instance, P(X < 1) given standard
normal distribution can be represented using Φ(1). Also Φ(2) = 98%,
Φ(1.65) = 95%.
‧Variance of Binomial distribution : np(1-p)
‧KL divergence:
┌ P(i) ┐
D_KL(P||Q) = Σ ln│ ─── │P(i)
i └ Q(i) ┘
1. Short Answer:
(a) What are PageRank and TFIDF? Why we need both of them to build a
good information retrieval model? (5 pts)
(b) Describe Central Limit Theory (5 pts)
(c) Describe p-value (5 pts)
(d) is KL-divergence a good 'distance' measure? If not, how to fix it
(5 pts)
2. 某次期末考有一百题“四选项”选择题，一位学生发现他对每一个题目确定正确解答的
机率只有0.6，如果他不确定的话，就任意猜一个答案。请问他答对超过80题的机率大
约是多少? (8 pts)
3. Let X be a uniform random variable on [0,1] and conditional on X=x let Y be
an exponential with parameter θ=1/x.
(a) Find the joint density of X and Y (5 pts)
(b) Find E[Y] (5 pts)
4. 某种灯泡已知寿命100小时，standard deviation 30 小时。台风天来了，假设现在我
　 们要库存一些灯泡，用以确保有98%以上的机率已有的灯泡连续点上可以发光至少2000
小时，那我们必须库存至少多少灯泡？ (8pts)
5. Let {X1, X2...Xn} be a sequence of Gamma random variables with parameter
α1,α2...αn and identical θ. Find the distribution function of ΣXi
i
-α
(Hint: the mgf of Gamma(α,θ) is (1-θ) ) (8 pts)
6. A statistics department at a large university maintains at tutoring service
for students. The hypothesis is that 40% of the students that using service
would be from bussiness department, 30% from engineering deparment, 20%
from social science department and 10% from agriculture. A random sample of
120 students revealed that 52, 38, 21, 9 students from each department,
respcetively. Using Chi-square test to check whether the sampling results
follow the hypothesis with α=0.05 (Chi-square table is in the end) (8 pts)
7. You want to build an automatic process to correct "Chiese sentences" (中式
　 英文) with misused verb. For example, you want the system to automatically
correct 'I opent the light' to 'I turn on the light'. You are given a large
set of English sentences written by native-Chinese writers, and a large set
of sentences from native English speakers. Please describe how you use
n-gram language model and noisy channel model to resolve this problem.
(10 pts)
8. If X1...Xn are iid and uniformly distributed between [θl,θu], please find
the maximum likehood estimation of θl and θu. (8 pts)
9. A coin (whose head-rate is p) is flipped until the first head occurs. Let X
denotes the number of flips Required,
(a) find the entropy H(X). (5 pts)
(b) Let Y denote the number of flips until the second head appears.
Thus, for example, Y=5 if the second head appears on the 5th flip.
Is H(Y) larger, smaller, or equal to 2H(X)? Explain your solution.
(5pts)
10. 大咩(Big Mia)、咩青(Mia green)、小羊(Little sheep)一起去一家有名的馒头店：
咩咩馒头店。咩咩馒头店有卖两种馒头，巧克力馒头和草莓馒头。巧克力馒头每十分
钟出炉一次，草莓馒头每八分钟出炉一次。大咩、咩青、小羊到的时候刚好馒头都卖
　　完了，只能等下一次的出炉。
　　小羊："草莓！草莓！我要吃草莓馒头（兴奋）。"
咩青："我要吃巧克力馒头"。
大咩看了一下时间，发现等一下就要上机率课了，于是说："我们还得赶去上机率课
　　才行，下一个出炉是哪个馒头，我们就全部吃那个馒头好了！！"
　　小羊和咩青："好吧～～"
　　问：
已知草莓馒头和巧克力馒头出炉的时间是互相独立的。
小羊成功吃到草莓馒头的机率是多少？（10 pts）　　
11. 在资工系举办了一个来测验同学们 coding 能力的竞赛，而最后进到决赛的有学生Ａ
和学生Ｂ，在决赛有着神奇的机器可以来测量学生的 "C Power"，就如同七龙珠的战
　　斗力探测器一般。已知学生Ａ每分钟可以产生100单位的 "C Power"，而变异数为24；
　　学生Ｂ则是每分钟可以产生80单位的 "C Power"，变异数为40。而在比赛剩最后10分
　　钟时，学生Ｂ暂时以40单位领先，请问到比赛结束结果是Ｂ产生较多的 "C Power"　
　　的机率为何？（10pts）
12. 在猎人试炼中，小杰和他的伙伴到达了最后一关。尼特罗会长（主考官）要求各位交
出自己的分数公式。而要成为一个顶尖的猎人，最重要的就是体力和智力。小杰的体
力是200（全部人的平均是150，变异数是40），智力为100（全部人的平均是130，变
　　异数是30），而我们知道体力和智力是独立的，所以小杰交出了公式为
Grade = 0.75 * Energy + 0.25 * Intelligence。
请问covariance(Energy, Grade)为多少？　（10pts）
Reference
Chi-square table:
┌─┬───┬───┬───┬───┬───┬───┬───┬───┐
│df│ 0.99│ 0.975│ 0.95│ 0.90│ 0.10│ 0.05│ 0.025│ 0.01│
├─┼───┼───┼───┼───┼───┼───┼───┼───┤
│ 1│ - │ 0.001│ 0.004│ 0.016│ 2.706│ 3.841│ 5.024│ 6.635│
├─┼───┼───┼───┼───┼───┼───┼───┼───┤
│ 2│ 0.020│ 0.051│ 0.103│ 0.211│ 4.605│ 5.991│ 7.378│ 9.210│
├─┼───┼───┼───┼───┼───┼───┼───┼───┤
│ 3│ 0.115│ 0.216│ 0.352│ 0.584│ 6.251│ 7.815│ 9.348│11.345│
├─┼───┼───┼───┼───┼───┼───┼───┼───┤
│ 4│ 0.297│ 0.484│ 0.711│ 1.064│ 7.779│ 9.488│11.143│13.277│
├─┼───┼───┼───┼───┼───┼───┼───┼───┤
│ 5│ 0.554│ 0.831│ 1.145│ 1.610│ 9.236│11.070│12.833│15.086│
└─┴───┴───┴───┴───┴───┴───┴───┴───┘