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.

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

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)

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

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

28. (5%) From the general knowledge, the population occurrence rate of asthma is 0.1 for children who visit hospitals. A pediatrician moved his clinic to a new place and he found 9 children with asthma among 50 appointments in his first day at the new place. If the 50 appointments are considered as Bernoulli random variables with O and 1 outcomes, the total number of asthma children, indicated as X, should follow a Binomial distribution with probability 0.1 under the general condition. He then makes the following guesses. Which one is a correct inference for the occurrence rate at the new place? (A) If the occurrence rate holds the same as other places, the standard deviation of each Bernoulli random variable in this problem is . The standard deviation of X should be 50 x 0.3=15 (B) Since 9/50=0.18 is greater than 0.1, there must be an increase of the occurrence rate (C) If the occurrence rate is greater than 0.1, we should also see the proportion of asthma children to be greater than 0.1 in the next day (D) Since P(X=9)=0.033 is smaller than 0.05, we will decide this as an unusual situation, and the evidence supports an increased occurrence rate. The threshold 0.05 can be applied when the number of appointments is 70 in a day (E) Since P(X≧9)=0.058 is greater than 0.05, we do not have to worry too much about the situation, and it might be a random fluctuation from day to day. The threshold 0.05 can be applied when the number of appointments is 70 in a day

29. (5%) A histogram of a dataset X is plotted in Figure (i). The 1st quartile, median and the 3'd quartile of the data are 10, 25, and 50. Which of the following statements is correct? i) (A) The truncated mean of the central 50% data should be (10+50)/2=30 (B) If another variable Y is positively correlated with X, the distribution of Y must be skew to the right as the distribution of X (C) The mode is greater than 25 (D) Since it is a skewed distribution, we can model the data with a x distribution (E) Suppose we randomly draw 30 samples with replacement from this dataset and take average to get X. The distribution of X is more likely to be the function f3(x) in Figure (ii) rather than the function f1(x) or f2(x) 110 130