課程名稱︰機率
課程性質︰資工系必修
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）︰2018/06/28
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%, mgf of a normal distribution is e^(μt+σ^2*t^2/2).
1. You want to estimate the size of an NTU class that is closed to visitors.
You know that the students are numbered from 1 to n, where n is the total
number of student.
You call three random students out of classroom and ask for their numbers,
which turn out to be 1, 3, 7. Find the maximum likelihood estimate for n.
(8pts)
2. the two random variables X and Y takes values in x∈{0,1} and y∈{0,1,2},
respectively. Their joint distribution function can be written as:
P(x,y) = K * (x+y)
Please calculate the joint entropy H(X,Y) and mutual information I(X;Y).
(12pts)
3. Suppose X follows N(0,1) and Y=B*X where B=2*(Bernoulli(0.5)-0.5)
(a) Show that Y follows N(0,1)
(b) Are X and Y independent?
(c) Calculate ρ(X,Y) (i.e. the correlation coefficient)
(15pts)
4. Suppose that 70% of the total 200 famalies in your neighborhood have
no dogs, 22% have 1 dog and 8% have 2 dogs, approximate the probability
that there are more than 90 dogs in your neighborhood. (8pts)
5. Suppose X1, X2, ... Xn are i.i.d with mean μ and variance σ^2,
let Yn = Σ(i=1~n) Xi, prove or disprove the following:
(a) E[Xi | Yn=y] = y/n
(b) E[Xi^2 | Yn=y] = y^2*(n^2-n+1) / n^2
(12pts)
6. Y=X1+X2 and Mgf(X1)=(e^t-1)/t, Mgf(X2)=t*e^(t/2)/(e^t-1)
(a) Use Chebyshef's inequality to find a lower bound of P(-2<Y<2)
(b) Do you think this lower bound attained from Chebyshef's inequality
is tight?
(12pts)
7. I want to test a hypothesis about the mean μ of a certain normal
population whose variance is known to be 4. My null hypothesis is
H0: μ=16 and my alternative is H1: μ≠16.
Initially I sample from the population 25 times.
(a) If I'm testing at 5% significance, what is the range of values
of X¯, my sample mean, that will lead to me accepting the null?
(b) Suppose tha the true mean is actually 20. What is the probability
that I will incorrectly accept the null?
(12pts)
8. Short Answer: (20pts)
(a) What are PageRank and TFIDF? Can you describe their pros and cons?
(b) What is KL-divergence? Is it a good 'distance' measure?
If not, how to fix it?
(c) What is noisy channel model? What kind of problem can be solved by it?
(d) Who invented Information Theory?
9. Let X be a discrete random variable. Show that the entropy of a function
of X is less than or equal to the entropy of X. (9pts)
10.Someone claims to have found a the last half of 紅樓夢 by 曹雪芹.
She asks you to decide whether or not the book was actually written by
曹雪芹. You buy a copy of 紅樓夢 and count the frequencies of certain
common words on some randomly seleted pages. You do the same thing for
the 'the last half'. You get the following talbe of counts.
Using this datamset up and evaluate a significance test of the claim
that the long lost book is by 曹雪芹. Use a significance level of 0.1.
(12pts)
Word 之 乎 者 也
紅樓夢 150 30 30 90
the last half 90 20 10 80