110年 - 110台灣聯合大學系統_碩士班招生考試_電機類:工程數學(B)#104927

科目：研究所、轉學考(插大)◆工程數學 | 年份：110年 | 選擇題數：2 | 申論題數：71

試卷資訊

所屬科目：研究所、轉學考(插大)◆工程數學

選擇題 (2)

Problem 4. Which of the following statement is NOT true? (A)

Problem 5. We toss two fair coins simultaneously and independently. If the outcomes of the two coins are the same, we win; otherwise we lose. Let A be the event that the first coin comes up heads, B be the event that the second coin comes up heads, and C be the event that we win. Which of the following statements is false? (A) Events A and B are independent. (B) Events A and C are not independent. (C) Events A and B are not conditionally independent given C. (D) The probability of winning is 1/2.

申論題 (71)

There are a total 22 problems. The first 3 problems are CALCULATION problems, and
the rest of 19 problems are MULTIPLE choices. All of the MUTIPLE choices problems
have only SINGLE correct answer. Please CLEARLY write your answers with associated
problem numbers on the answer sheet. There will be NO partial credit given to the MUL-
TIPLE CHOICE problems, anything that is not on the answer sheet will NOT be counted.
Problem 1. Given a matrix

where a is a real number. Find the rank of A| and 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 matrix

in the Jordon form, i.e.,

(1) Find the range space (i.e., column space) of A, denoted as R(A), and the row space of A (i.e., R(

denotes the transpose of A;

(2) find the projection matrix

(3) find the eignevalues and eigenvectors of

(1) Is Tr(AX) ≥ 0 true?

(2) Is it true that if Tr(AX) = 0, then AX = 0

(zero matrix)? Prove or disprove your answer.

(i) always

(ii) sometimes

(iii) never independent.

(i) X and Y can be independent or dependent depending upon the strictly positive values

(ii) X and Y are always independent

(iii) X and Y can never be independent

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

(i) EY] =1

(ii) EY] =2

(iii) E(Y]=0

(i) E[X₁|X=2]=0.1

(ii) E[X₁ I X= 2]=0.5

(ili) E[X₁| X=2]=0.25

(i) 1/6

(ii) 1/3

(iii) 1/2

(iv) 1/4

(i) E[XY] = E[X]E[Y]

(ii) E(X² + Y²] = E(X²] + E[Y²]

(iii)fxtr(x+y)=fx(x)fr(y)

(iv) var(X+Y)=var(X)+uar(Y)

(i) fz(4.5)=0

(ii) fz(4.5)=1/8

(iii) fz(4.5) = 1/4

(iv) fz(4.5) = 1/2

(i) 0

(ii) 9/4

(iii) 3/4

(iv) 1/4

(i) λ

(ii)

(iii)

(iv)1

(i) W is Poisson with parameter min(λ₁, λ₂)

(ii) W is Poisson with parameter λ₁ + λ₂

(ii) W may not be Poisson but has mean equal to min(λ₁, λ₂)

(iv) W may not be Poisson but has mean equal to λ₁ + λ₂

(i) P(X =0) =P(X = 50)

(ii) P(X = 51) ＞0

(iii)

(iv) P(X = 50) = 0.6

(i)1-Φ(5)

(ii) Φ(5)

(ii)

(iv)

(i)

(ii)

(iii)

(iv)

(i) On average, you will lose money to your smart friend

(ii) On average, you will lose neither lose nor win. That is, your average gain/loss is 0.

(iii) On average, you will make money from your friend