課程名稱︰機率
課程性質︰資訊系必修
課程教師︰林守德
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰2017/6/22
考試時限（分鐘）：180
試題 :
Total Points: 120
You can answer in either Chinese or English.
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%.
1. Short Answers: (20pts)
(a) Can you describe how noisy channel model can be used for machine
translation between English and Chinese?
(b) Describing two key approaches for information retrieval.
(c) What is KL-divergence? How to make such measurement symmetric?
(d) What is the relationship between MI(X,Y) and H(X|Y)
2. A fair coin is flipped until first head occurs. Let X denote the number
of flips required. Find the entropy H(X). (8pts)
3. Suppose that we roll a standard fair 6-side die 100 times. Let X be the
sum of the numbers that appear over the 100 rolls.
What is Pr(|X-350|>50)? (8pts)
4. The average age of major league baseball players is 28.15 years with a
standard deviation 2.3 years. If we can assume that ages are Normally
distributed, what is the probability that the average age of 10 randomly
selected Red Sox players is less than 27 years? (8pts)
5. Let the random variables X and Y have the joint p.m.f f(x,y) = c(x+y);
x = 0, 1, 2; y = 0, 1 with y < x.
Find the correlation coefficient ρ (10pts)
XY
6. In a pseudo-world, there are only two types of blood: 40%type A, and
60%type B. The King randomly chooses 200 couples and asks their blood
type. It turns out that 40 of them are A-A, 80 of them are A-B, and 80
of them are B-B. Assume that H means that the marriage of people has
0
nothing to do with their blood type (i.e. independent). Do you have
95% confidence to reject H ? Please explain your answer. (10pts)
0
7. Given a sequence of independent, indentically distributed random variables
X : n = 1, 2, ... such that P(X = -1) = p and P(X = 1) = 1-p.
n n n
Let Y = X * X ; n = 1, 2, ...
n n n+1
Show that sequence of Y doesn't follow the Markov Property if p!=1/2 (10pts)
8. Let X be a uniform random variable on [0,1], let Y be an exponential
distribution with parameter λ = 1/x given X = x, Find E[Y]. (8pts)
9. The life of a kind of hard drive has mean 100 hr and standard deviation
30. We have a very important PC that utilizes this hard drive and we need
it to last for at least 2000 hrs with 98% confidence. How many hard drives
we need to prepare for replacement for such purpose? (8 pts)
10. If X ...X are iid and uniformly distributed between [θ ,θ ],
1 n l u
please find the maximum likelihood estimation of θ and θ .
l u
11. You want to use reinforcement learning to build a self-driving car.
Define the essentials of reinforcement learning components (e.g. reward,
policy, state) to solve this task. (10 pts)
12. John builds an auto-translation tool, and wants to claim that the tool
he builds is better than the Google Translator (whose accuracy is 75%).
He tested on 100 sentences and found 76 of them were translated correctly.
(a) What is the 95% confidence interval of the accuracy of his system
(hint: Z = 1.65, Z = 1.96)? Can you use confidence interval to
0.05 0.025
convince people John's system is better? (5pts)
(b) Given H = 75%, what is the p-value? You don't need to calculate
0
the exact p-value, just list its equation is good enough. (5pts)
(Chi-square table @ https://people.richland.edu/james/lecture/m170/tbl-chi.html)