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 12. (25 %) Let Xi, 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 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 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