16. (5%) According to Problem 15 above, for the 3✖3 covariance matrix of (X₁.X₂,X₃), determine the correct element values Ki of the covariance matrix.
(A)
(B)
(C)
(D)
(E) None ofthe above.

複選題十五. Consider the famous hat problemn. Suppose that n pcople throw their hats in a box and n each picks one hat at random. (Each hat can be picked by only one person, and cach assi of hats to persons is equally likely) Let X be a random variable that ropresents the number of people that get back their own hat. Which of the following statemnents is/are true? (A) When n =3, P(X =3)= (B) When n = 3, P(X = 1)= (C) When n = 4, P(X = 3) = 0. (D) When n = 4, E[X] = 2. (E) When n = 5, E[X] = 1.

複選題十六. Consider the famous hat problem as in the previous problem. Let X, be a random variable that takes value 1 if the ith person selects his/her own hat, and takes value O otherwise. Let cov(Y, Z) = [Y . Z] -E[Y] ㆍE[Z] be the covariance of two random variables Y and 2. Which of the following statements is/are true? (A) When n = 4, cov(X4, X4) is .(B)When n = 4,cov(X1,X2)=(C) When n = 4, va=2. (D) When n = 4, Xi and Xz are statistically independent. (E) For any two random variables Y and Z, cov(Y, Z) ≥0.

X and Y are independent continuous random variables with probability density functions (PDFs) shown below. What is the probability of P(-1

15. (5%) The random variables X1, X2,X3 have the joint Gaussian pdf: What is the marginal Goint) pdf of Xi and X3? (A) 0. (B) (C) (D)(E) None of the above. 

1. If , the solution set may be a subspace. (A) True. (B) False.

2. If , the solution set is not a subspace. (A) True. (B) False.

3. If , the solution set may be or may not be a subspace. (A) True. (B) False.

4. If , the solution set is not a subspace. (A) True. (B) False.

5 . If , the solution set is exactly a subspace. (A) True. (B) False

複選題 Which of the following is a vector space?(A) P={p|p is a polynomial in of any order}(B)={p|pis a polynomial in of exactly th order}(C){T| T： is a linear transformation}(D) The set of all matrices of which the sum of all entries is zero, i.e., .(E) The set of all square matrices with trace 1 , i.e., .