[試題] 106-2 林守德機率期末考

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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 --

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