Problem 13. (2.5 %o) Let X₁, X₂, X₃ be independent random variables with the continuous distribution over [0,1]. Then P(X₁＜X₂＜X₃)=
(A) 1/6
(B) 1/3
(C) 1/2
(D) 1/4

Problem 14. (2.5 %) Let X and Y be two continuous random variables. Then, (A) E[XY] = E[X]E[Y](B) E[X2 + Y2] = E[X2] + E[Y2](C)fX+Y(x+y)=fX(x)fY(y)(D) var(X+Y)=var(X)+var(Y)

Problem 15. (2.5 %) Suppose X is uniformly distributed over [0,4] and Y is uniformly distributed over [0,1]. Assume X and Y are independent. Let Z = X+Y. Then (A) fz(4.5) = 0(B) fz(4.5)=1/8 (C)fz(4.5)= 1/4(D) fz(4.5)=1/2

Problem 16. (2.5 %) For the random variables defined in problem 13, P(max(X,Y)＞3) is equal to (A) 0(B) 9/4(C) 3/4(D) 1/4

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 19. (3.0 %o) Let X be a random variable whose transform is given by MX(s) = (0.4 +0.6es)50. Then, (A) P(X =0) = P(X = 50) (B) P(X = 51) ＞0 (C) P(X = 0)=0.450 (D) P(X= 50) = 0.6

Problem 20. (3.0 %) Let Xi,i = 1,2, ... be independent random variables all distributed according to the pdf fX(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 Xi,i = 1,2, ... be independent random variables all distributed according to the pdf fX(x) = 1 for 0 ≤ x ≤ 1. Definc Yn = X1X2X3... Xn for some integer n. Then var(Yn) is equal to (A)(B)(C)(D)

Problem 22. (3.0 %) Suppose you play a matching coin game with your friend as follows. Both you and your friend have a coin. Each time, you two reveal a side (i.e. H or T) of your coin to each other simultaneously. If the sides match, you WIN a 1 from your friend if sides do not match then you lose a 1 your friend. You decide to go with the following strategy. Every time, you will toss your unbiased coin independently of everything else, and you will reveal its outcome to your friend (your friend does not know the outcome of your random toss until you reveal it). Then, (A) On average, you will lose money to your smart friend (B) On average, you will lose neither lose nor win. That is, your average gain/loss is 0. (C) On average, you will make money from your friend

1.國防政策可以反映國家國防體制運作的目的與取向，另一方面還可以透過國防政策的單一化，爭取國人認同，擴大國防政策的效益。(A)O(B)X

110年 - 110 國立中央大學_碩士班招生考試_機械工程學系、機械工程學系(光機電工程)、能源工程研究所_(共同)：工程數學#125285

110年 - 110 國立臺灣大學_碩士班招生考試_部分系所：工程數學(A)#125255

110年 - 110 台灣聯合大學系統_碩士班招生考試_電機科:工程數學(C)#125211

110年 - 110 台灣聯合大學_碩士班招生_電機類：工程數學C#124898

110年 - 台聯大_工程數學B#113346

110年 - 110 國立臺灣科技大學_碩士班招生試題_材料科學與工程系(乙組)：工程數學#113288

110年 - 110 國立台北科技大學碩士班招生考試_自動化科技研究所：工程數學#111318

110年 - 110 國立臺灣大學_碩士班招生考試_部分系所:工程數學(D)(含線性代數、機率與統計)#111303

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A#110408

110年 - 110 國立臺北科技大學_碩士班招生考試_機械工程系機電整合碩士班：工程數學#107430