110年 - 台聯大_工程數學B#113346

Problem 4. (2.5 %) Which of the following statement is NOT true? (A) If A ⊂ B, then P(A）≤ P(B）. (B) If P(B）＞ 0, then P(A|B) ≥ P(A）. (C) P(A∩B) ≥ P(A）+P(B） - 1. (D) P(A∩B^c) = P(AUB) - P(B）.

Problem 6. (2.5 %o) Suppose X, Y and Z are three independent discrete ran- dom variables. Then X and Y + Z are. _____ independent. (A) always(B) sometimes (C) never 

Problem 7. (2.5 %) Consider two random variables X and Y, each taking values in {1,2,3}. Let their joint PMF be such that for any 1 ≤ x,y ≤ 3, 
Then, (A) X and Y can be independent or dependent depending upon the strictly positive values (B) X and Y are always independent (C) X and Y can never be independent

Problem 8. (2.5 %o) We throw n identical balls into m urns at random, where each urn is equally likely and each throw is independent of any other throw. What is the probability that i-th urn is empty? (A) (B)(C) (D)

Problem 9. (2.5 %) For a biased coin, the probability of 'heads" is 1/3. Let h be the number of heads in five independent coin tosses. What is the probability P(first toss is a head | h = 1 or h = 5)?(A) (B)(C) (D)

Problem 10. (2.5 %) A well-shuffled deck of 52 cards is dealt evenly to two players (26 cards each). What is the probability that player 1 gets all the cases (A) (B)(C)(D)

Problem 12. (25 %) Let X_i, 1 ≤ i ＜ 4 be independent Bernoulli random variable each with mean p = 0.1. Let X = That is, X is a Binomial random variable with parameters n = 4 and p = 0.1. Then, (A)E[X₁ | X=2]=0.1(B)E[X₁ | X= 2]=0.5(C)[X₁ | X=2]= 0.25

Problem 17. (2.5 %) N people put their hats in a closet at the start of a party, where each hat is uniquely identifed. At the end of the party each person ran- domly selects a hat from the closet. Suppose N is a Poisson random variable with parameter. If X is the number of people who pick their own hats, then E[X] is equal to (A) (B) (C) (D)1

Problem 18. (3.0 %) Suppose X and Y are Poisson random variables with parameters respectively, where X and Y are independent. Define W = X + Y, then, (A) W is Poisson with parameter (B) W is Poisson with parameter (C) W may not be Poisson but has mean equal to(D) W may not be Poisson but has mean equal to

Problem 20. (3.0 %) Let X_i,i = 1,2, ... be independent random variables all distributed according to the pdf f_X(x) = x/8 for 0 ≤ x ≤ 4. Let S = . Then P(S ＞3) is approximately equal to (A) (B)(C) (D)

Problem 21. (3.0 %) Let X_i,i = 1,2, ... be independent random variables all distributed according to the pdf f_X(x) = 1 for 0 ≤ x ≤ 1. Definc Y_n = X₁X₂X₃... X_n for some integer n. Then var(Y_n) is equal to (A)(B)(C)(D)

Problem 1. (10 %) Given a matrix

where a is a real number. Find the rank ofand give the corresponding range of a.

(i) (10 %) Find the Eigenvalues and Eigenvectors for .

(ii) (5 %) Show that there exists a matrix, which is similar to the matrixin the Jordon form, i.e., 

(i) (15 %) Letwith r singular valuesU_r= [u₁,⋯,u_r]
consisting of the associated r left singular vectors and V_r= [v₁,⋯,v_r]
consisting of the associated r right singular vectors.
(1) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R(A^T)) where A^T denotes the transpose of A ；
(2) find the projection matrix P_A such thatand；
(3) find the eignevalues and eigenvectors of AA^T.

(ii) (10 %) Suppose that A,
and Tr(A) denotes the trace of A (i.e., the sum of all the diagonal elements of A).
(1)Is Tr(AX)≥0 true?
(2)Is it true that if Tr(AX) = 0, then(zero matrix)?
Prove or disprove your answer.