15.(5%) Let  Which value of a can make A defective? 
(A) No value of a can make A defective
(B) -2
(C) 2
(D) 0 6 1 10
(E) 1

11. (5%) Suppose that the characteristic polynomial of a matrixA is p(λ)=λ(λ-1)(λ-2)2(λ+3)3. Which of the following statements about the matrix A is correct? (A) The size of A is 4 X 4 (B) A is invertible (C) det( A ) =-128 (D) (2I -A)x= 0 has nontrivial solutions (E) A has nullity O

12. (5%) Let A be an n by n matrix. Which of the following statement is NOT identical to others? (A) Ax = 0 has a nontrivial solution (B) The rank of A is less than n (C) The dimension of A's null space is positive (D) A is not diagonalizable (E) The singular values of A are not all positive 乙L

13. (5%) Let vi, ... , Vn be vectors in a vector space V for n ＞ 0.Which statement is NO'T correct? (A) If the dimension of V is n, and vi, ... , In span V, then yi, ... , vn are linearly independent (B) If the dimension of V is m, and m ＞ n, then vi, ... , vn can be extended to form a basis for V (C) If the dimension of V is m, and m ＜ n, then vi, ... , In are linearly dependent (D) If both fu, ... , Ms and fu, ... , um； are bases for a vector space V, then n = m (E) A vector v in Span(vI, v) can be written uniquely as a linear combination of VI, ... , In if and only if v, , vn are linearly independent [4 7 37

14.(5%) Which one is NOT an eigenvalue of (A)1 (B) 2 (C) 3 (D) 4 (E) 5 O I 4 -2]

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 

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)