21.(5%) Y1, Y2, ... are independent and identically distributed random variables； the moment generating function of each of them is Let X= Find this probability mass function of X, px(X)

17. (5%) Let X and Y be independent random points from the interval (-1,1). Find E(max(X,Y)). (A) 2/3 (B) 1/2 (C) 1/3 (D) 1/4 (E)0

18.(5%) A stick of length 1 is broken into two pieces at a random point. Find the correlation coefficient and the covariance of these pieces. (A)1/2 (B) -1/2 (C) 1/12 (D)-1/12 (E) 0

19. (5%) A traffic light can be in one of the three states: Green (G), Red (R), and Yellow (Y). The light changes in a random fashion. At any one time, the light can be in only one state. The experiment consists of observing the state of the light. Let a random variable X(') be defined as follows: X(G)=-1, X(R)=0, X(Y) = π； assume that P(G) = P(Y) = 0.5 P(R), what is P(X ≦ 3)? (A)1/4 (B) 1/2 (C) 3/4 (D)1 (E) 0 

20.(5%) Let X1,X2, ....Xn be n identical and independently distributed exponential random variables with fxi(x) = e-xu(x), u(x) = 1 for x ≧ 0,0 otherwise. The maximum value for the probability density function, fzn(z), of the random variable Zn = max(Xi, X2,.... Xn) occurs at z =? (A) In(n) (B) In(-1) (C) In(1/n) (D) (1 -e-1)n (E) n(1-e-1)n

22. (5%) A newly married couple decides to continue having children until they have one of each sex. If the probability of a child being a boy is 1/4 and the probability of a child being a girl is 3/4, how many children should this couple expect? (A)4(B)17/4(C)13/3(D)9/2(E)16/3

23.(5%) The two random variables, X and Y, are independent and exponentially X distributed, each with mean , Find Fz(z), which is the cumulative distribution function of Z.

24.(5%) We have some earthquake data that occurred in a region over years, summarized in the following table: Assume that earthquakes are independent rare events and their occurrences are homogeneous over time. Under these assumptions, the number of earthquake events follows Poisson distribution. Estimate the probability of having earthquakes with magnitude M 2 5 in the next month. (A)3.8% (B)21.3% (C) 48.7% (D)66.7% (E) unable to evaluate 

25.(5%) Which of the following statements are correct? (1) Statistical independence implies no correlation (2) No correlation implies statistical independence (3) Statistical independence and no correlation are equivalent (4) In a simple linear regression problem, a large value of the least square estimate for the slope coefficient implies a strong association between the response and the explanatory variables (A)(1) (B)(2) (C)(3) (D)(1)(4) (E)(1)(2)(3)(4) 

26. (5%) Hospital A is experimenting with an AI-assisted system for judging the status (+ or -) of a certain disease. A pilot study is conducted to compare the results assessed by the AI system and a trained doctor. 50 patients are recruited, and the results are reported in the following table: The goal of the study is to verify if the assessment from the AI system is consistent with that from a trained doctor. Which of the following statistical test can be used to answer the question? (A) 5 independent 2 sample t-test (B) paired 2 sample t-test (C) chi-squared test (D) signed-rank test (E) F test

27. (5%) Based on the playing history of Team A and Team B, Team A has a 60% chance of winning basketball in one game. Assume that the outcomes of their games are independent. Which of the following statements is correct? (A) Team A has a better chance of winning in 2-out-of-3 gameplay(三戰兩勝) rather than in 3-out-of-5 gameplay(五戰三勝) (B) Team A has a more than 70% chance of winning in 2-out-of-3 gameplay (C) For 2-out-of-3 gameplay, the chance of playing the third game is more than 40% (D) Given that the third game takes place in 2-out-of-3 gameplay, Team B has a less than 40% chance of winning the game (E) None of the above