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

16. (5%) Kim is at a train station, waiting to make a phone call. Two public telephone booths, next to each other, are occupied by two callers, and 1 1 persons are waiting in a single line ahead of Kim to call. If the duration of each telephone call is an independent exponential random variable with λ= 1/3, what is the variance of Kim's waiting time?Kim's waiting time is defined as the time interval from now to the moment that a phone is available for Kim to make his call. (A) 18 (B) 27 (C) 99/4 (D) 99 (E) 108

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 

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

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