課程名稱︰機率 課程性質︰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）︰2013/06/17 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Probability 2013 Final (Prof. Shou-de Lin) 6/17/2013 14:30-17:30 Total Points:120 You can answer in either Chinese or English. 1. Short Answer: (a) What are PageRank and TFIDF? Why we need both of them to build a good information retrieval model? (5pts) (b) draw the Venn Diagram of Entropy and Mutual Information of two random variables (5pts) (c) What is noisy channel model? What kind of problem can be solved by it? (5pts) (d) Is KL-divergence a good 'distance' measure? If not, how to fix it?(5pts) 2. (revisit midterm II) The probabilities of failure on component 1 (follow Poisson of probability λ1=4) and component 2 (follow Poisson of probability λ2=5) of a product are independent. We observe 3 product failures, what is the probability that two of them come from component 1? (6pts) 3. (revisit midterm II) Let X and Y be two independent random points from the interval (0,1). Calculate the CDF of min(X,Y)/max(X,Y) (8pts) 4. (revisit midterm II) Let {X1, X2,...,Xn} be a sequence of Gamma random variables with parameter α1, α2,...,αn and identical θ. Find the distribution function of ΣiXi (hint: the mgf of Gamma(α, θ) is (1-θt)^-α) (8pts) 5. (revisit midterm II) A point (X,Y) is selected randomly from the triangle with vertices (0, 0), (1, 1) ad (1, 0). Calculate E(X|Y=y) (6pts) 6. In the class, we have discussed the n-gram language model. Indeed there is a tradeoff in determining n: larger n allows us to model the longer-distance dependency among words without significantly increasing the storage size? (10pts) 7. 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. (8pts) 8. A playoff consists of a three-game series that terminates as soon as either team wins two games. Let X be the random variable that represents the outcome of a playoff between teams A and B; examples of possible values of X are AA, and BAB. Let Y be the number of games played, which ranges from 2 to 3. (a) Assuming that A and B are equally matched and that the games are independent, calculate H(X), H(Y), H(Y|X), and H(X|Y) (10pts) (b) Let Z denote the winning team. Find H(X|Z) (5pts) 9. A certain geneticist is interested in the proportion of males and females in the population that have a certain minor blood disorder. In a random sample of 1000 males, 250 are found to be afflicted, whereas 275 of 1000 females tested appear to have the disorder. Compute a 95% confidence interval for the difference between the proportion of males and females that have the blood disorder. (hint: the variance of binomial is np(1-p)) (8pts) 10.Suppose that X and Y are independent exponential random variables with parameter λ, and let Z = X/(X+Y), show that Z is a uniform distribution over (0,1) (8pts) 11.A simple random sample of 1000 eligible voters was chosen to study the relationship between sex and participation in an election. The results are summarized in the following table (8pts): Men Women Vote 260 360 Didn't vote 140 240 12.Let {X1, X2,...,Xn} are integers, generated by random round-off sampling from a uniform distribution U[0, b], Now we want to estimate the parameters b, this is known as German tank problem in WWII. (Sample are the German tank serial numbers spot by the Allies, and we want to estimate total number of tanks German has) (5*3=15pts) a. Find the estimation of parameters b using Maximum Likelihood Estimation. b. Find the estimation of parameters b using Method of Moments. c. What are the potential concerns for each of the estimation? Reference KL divergence: 略 Poisson distribution: 略 Chi-square table: 略 --

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